Let us say we will begin our instruction today. So, start with the review as usual. So, last day that means day before yesterday, what we did was to discuss first the Bayes theorem famously invented by Thomas Bayes some more than 200 years back and I have told you that application of Bayes theorem or the whole Bayesian domain of analysis in statistics and probabilities a lot more than what we had as Bayes theorem. And if you have checked your Moodle page of this course, then you might have seen that I have uploaded one or two links for additional readings for fun. You can get to know more about what Bayes theorem is, what are the different application of Bayes theorem is. It is a lot wide and it is amazing I mean where it has reached over the last 200 years, not actually last 200 years I mean probably within last 100 year or so, because after Bayes had developed the theorem, nobody talked about it for a while then it started again. So, we have gone through one or two examples on Bayes theorem. Then we discussed random variables, we discussed their definitions and then we discussed basic functions of random variables. For discrete random variables and continuous random variables, we have functions like the probability mass function, cumulative distribution function and probability density function and also discuss their properties with some examples. Finally, we just touched the joint distribution part with an example for joint probability mass function. Today's lecture starts with the definition of expected value of a random variable which is denoted by this notation E within square bracket x, x is the random variable. The definition given here is for the discrete one where you define it as the summation of this quantity x i times the this is the probability mass function. If you remember small p stands for mass function capital X in the subscript is the random variable under consideration and x i is the value that the random variable takes. So, if we sum it up over all the i values then we get the expected value of that random variable. This is how we define it for a discrete random variable and this is the definition for a continuous random variable where we have the same function x times instead of mass function now we have the density function and we instead of summing it up we integrate it over the whole range that means minus infinity to plus infinity. So, basically if you see their same formulas one applies for the discrete case, one applies for the continuous case. What it is basically if you look at it in detail is the weighted average of all the values that the random variable x can take. Where is the weight? This is the weight. So, this is basically nothing but the frequency it will more clear if you look at the discrete case. So, this is the frequency mass function is equivalent to frequency. So, we are weighting the value that x is taking x i by the frequency and you are taking a weighted average. So, the expected value is nothing but the weighted average of all the values that the random variable x can take. It holds good for both discrete and continuous random variables. This quantity the expected value of x is also known as the first moment of x. We will see a little more about that later. As an example I cite a very simple one for a discrete case.  So, this is the value 1. What is the likelihood of that? What is the PMF probability mass function of that? It is 1 over 6 because all the numbers from 1 to 6 are equally likely. So, 1 times is PMF, 2 times the PMF, 3 times the PMF that is wrong it should be 6 times the PMF. If you add them all up you get 3.5. So, 3.5 is the expected value of the variable x where x is the outcome of rolling a fair six sided die. So, the question is it same as the mean of the random variable? Is expected value is same as the mean of the random variable? If you look at this example of course, you have this six numbers 1, 2, 3 and 4, 5, 6 the range is from 1 to 6 all of them are equally likely. So, you can easily see that 3.5 is nothing but the mean. Also if you look at the previous definition that it is nothing but the weighted average of all the numbers. Then also you can say that this is the mean of the random variable. We have discussed about mean earlier. So, I am going to what mean is anyway. So, we have a new name for mean which we call now expected value when we are talking about random variables. We can also define expected value of a function of x. So, g x over here is a function of x and we can obtain the expected value of that function of x as g x multiplied by again the mass function that is the weight and integrate it over minus infinity plus infinity. So, this is the definition for a continuous random variable. Similarly, you can have also a definition for the discrete random variable and you can obtain g x i multiplied by the probability mass function. Sum it up and you get the expected value of a discrete random variables function. So, this should be g x i will make that correction. So, this is e g x just like this one and I told you about moment. We said that the expected value is also known as the first moment. Expected value of x is the first moment of x. Here we discuss about the nth moment of x which is defined as expected value of x raised to n and how do you obtain this? Of course, x raised to n is a function of x. So, we write that function of x times the pdf of x and we integrate it from minus infinity to plus infinity. That means the whole range and you obtain the expected value of x raised to n and this is known as the nth moment of x. So, automatically if you put n equal to 1, you get the first moment of x and if you put n equal to 2 as you have put over here, we get the second moment of x and so on. It is clear? Many of you are familiar with these things, but this definition of moment I want to clarify again. Many a times people assign these terms moment of x or so to various other functions. You need to understand the differences. Basically, moment of x is the expected value of x raised to n that is the nth moment of x. Sometimes people also assign this name to a function of expected value of x n which is not really true, but you can say that it is order, it is of that order of that order of the nth moment of x or so. I will come back to that. The other very important property of a random variable distribution is its variance and how do you define variance is denoted as var and x within a square bracket is defined as the expected value of the quantity x minus mu x. What is mu x? What does this symbol define mu x is? The mean of the random variable x. As you can also see that x in the subscript is written in capital letters which means that x is the random variable and mu denotes its mean. So, x minus mu x square, you are taking an expected value of it which we call the variance of x. You can also write that variance of x is nothing but expected value of x square minus square of the mean. We will see later on how these two things are same. So, expected value is same as the mean we have studied that earlier and we know that mean is a measure of central tendency of a distribution. The variance on the other hand is a measure of the dispersion or spread that you can have in a distribution or for a random variable. So, two measures first two measures one the first one that is very important is of the central tendency then we go for the measurement of dispersion or spread in the distribution of a random variable. Do you know what are the next measures of distributions? Have you heard any terms? Did you hear skewness? Did professor Condon talk to you about skewness and kurtosis? So, those are the next level or next higher order of defining the spread of a random variable or as the spread of a distribution. First one the central tendency of a random variable then the dispersion random variable then you go for those two skewness and kurtosis and each one we go we go one level higher in the moment of x. You have seen that the expected value is the first moment of x variance is here you can see is a function of the second moment of x and so on. So, that is what we say over here that variance is a function of second moment of x. Many a times we just wrongly say that the variance is the second moment of x it is not really true, but well you can say that variance is also a measure of the dispersion or spread in x. The same parameter has another notation it is sigma square variance is also denoted by sigma square and with a subscript x x written in capital to denote that it is the variance of the random variable x. And we will also see that the square root of this quantity this variance is sigma x where sigma x is nothing, but the standard deviation of the random variable x. So, although this symbol does not necessarily mean that it is square of this quantity, but it actually is, but we will better look at this symbol as sigma square not as square of sigma is that it is like just like any other symbol like variance and square brackets there you have a symbol of sigma square not necessarily square of sigma although is the same quantity. So, we come back to the two different definitions for variance we said that those two are equal and we will see how. So, you are trying to equate these two expected value of x minus mu whole square and expected value of x square minus mu square. So, I have just dropped the subscript x, but mu here stands for mu x for this case because there is no other random variable this is how it goes. So, what we do first we expand that parenthesis x minus mu square we get x square minus 2 mu x plus mu square. And then we find the expected value of each one separately. So, expected value of x square expected value of 2 mu x expected value of mu square. In the next step we have this then you take the mu out of the expectation based on what consideration the mean or the expected value same thing of a random variable is not a random variable the mean of the random variable is a constant or deterministic value. So, what is its expected value its value does not change. So, the expected value of mu is same as mu expected value of any constant which is not a variable is the same constant. So, you can take that constant out of the parenthesis here that is what you are doing. So, we take 2 and mu out of the parenthesis we take mu square out of the parenthesis because those two are deterministic quantities not random variables. So, then you simplify we get this which also means mu is nothing but the expected value of x. So, you can also write it in this way very simple proof, but it gives you the idea of what how the operator expected value works on a random variable how it works on a constant value. We will see more of that when you discuss about the properties of expected values and variances.  So, we first find out see it involves finding out the expected value of the x square and this one of course, we know already. So, the second moment of x or expected value of x square we obtain by x square times the mass function x square times mass function. So, on again this should be 6 not 7 I am sorry for this this is what we obtain as the you can say mean of x square or expected value of x square. So, the variance of x is this value 91 over 6 minus the mean that you have already obtained. So, this is what you get as the variance of that. So, I gave a 2 examples of obtaining expected value and variance for discrete random variable. Can you repeat that for a continuous random variable? Very simply we just have to replace the probability mass function by probability density function and the summation by an integral minus infinity to plus infinity ok. So, process thus will come next and we will talk more about continuous random variables. So, I will try to give my examples more on the discrete side. Then we come to the properties of these two quantities the expected value and variance of random variable x. So, if we have these two constants a and b constants means they are not random variables they are deterministic quantities we know their value exactly what they are. So, for that if you want to evaluate the expected value of a x plus b what is this? This is the function of x right function of the random variable x and how can you obtain this? You can obtain it by the formulation of if it is a discrete random variable then you can write g x times mass function of x sum it up over all i values. If it is a continuous random variable then you can write the function times the pdf of x and integrate it over minus infinity to plus infinity range ok. So, what you get is a times expected value of x plus b why because the constant value has an expected value equal to the same value right. So, you can take it out and you can take this out ok. Just like we did for the previous example where we tried to equate the two defined two different definitions of the variance. And if you are trying to find out the expected value of two different random variables x and y then we do the same thing it will be expected value of x plus expected value of y ok. X minus y again expected value of x minus expected value of y we will see a more general definition for this. Just a simple proof of y e x plus y will be equal to e x plus e y. So, we write e x plus y as this. So, here you can see although I did not specifically define it over here how do you define the expected value for a function of two different random variables. We write the function like g x y times the pdf. Now, instead of a pdf we have a joint pdf. Remember we discussed joint distributions in the last class. So, we have a joint pdf evaluated at capital X equal to small x and capital O equal to small y. And we integrate it over both x and y for the whole range of x and y that means minus infinity to plus infinity. Is that definition do you see the similarity with the previous one? Very simple just replace the single pdf with a joint pdf. So, now we break it up x times the joint pdf and y times the joint pdf two different integrals. The first one gives us expected value of x the second one gives us expected value of y. How? If you integrate this with respect to y what do you have? If you integrate y from minus infinity to plus infinity do you remember that example that we discussed on the joint probability mass function that you if you integrate on the other one then you get the individual density or mass function for x. If you integrate over x for the whole range of x you get the density or mass function of y. So, just considering that this one will reduce to single integration x f x x. This one will be reduced to single integration y f y y. Finally, you will have expected value of x plus expected value of y and this is the more general definition. If we have a function of multiple random variables. So, y is a function of random variable not one, but many. What you have over here are random variables x 1 to x n and we have a linear combination. So, a is here is a constant plus a 1 another constant times random variable x 1 a 2 another constant time random variable x 2 and so on up to a n x n and the expected value for this is expected value of y equal to a 0 plus a 1 times expected value of x 1 and so on up to a n times expected value of x n. Does that make sense to you based on what you discussed just now and previously like expected value of x plus y is expected value of x plus expected value of y. We are using that also earlier we said that expected value of a x plus b where a and b are constants is x a times expected value of x plus b. So, you can take the constants out of the expectation and you have a linear combination of the expectations. So, this is what we have a i expected value of x i summed up from 1 to n. Now, if you look at the variance the variance of a x plus b is a square times variance of x. And the b the other constant the lone constant which was not associated with random variable does not appear in the variance. So, that lone constant does not change the variance why is it any idea. When we subtract the mean that b constant will also appear. So, it will get subtracted. Subtract the b from that in the expression for variance. In the definition for expression for variance you have a x square minus mu square. Mu square. So, there the b will cancel out. I would like you to have a more physical feeling of that. So, when we discuss the continuous distributions like normal and normal then you will see how that constant not associated with any random variable does not really affect the variance in the system. Again we take that general linear combination of random variables y equal to a 0 plus a 1 x to a n x 1. For that the variance we have is the summation of a i square variance of x i. So, variance is multiplied by a i square not a i mean was multiplied by a i and this constant a 0 does not appear in the variance. Same thing it is a second moment. So, that is why you have square and of course, this one cancels out just like your friend said. We have another measure of dispersion. It is a relative measure of dispersion. We use it often times where we define the coefficient of variation v with a subscript x again x in capital because it knows the random variable x define as the ratio of the standard deviation to mean of the random variable. And you can express it sometimes in terms of percentage and usually it gives a better idea of the spread usually why? Let me give you an example. Let us say you are you are you are you are considering two random variables comparing that is why you use it more often use this coefficient of variation more often. You are considering two random variables like age of the faculty members one random variable and age of the students both in IIT other random variable. Let us say the mean somewhere around for the first one 40 and for the second one 20. Now if you count standard deviations for the first one you have 5 for the second one you have 3. So, now if you consider absolutely the standard deviations you see that the standard deviation is more for the variable age of faculty in absolute term 5 greater than 3. But if you are considering in relative terms you have 5 over 40 for the faculty's age and 3 over 20 for the students age. And I would say that there is more spread in the variable students age. Do you see my point? Do you agree to what I am saying? Most spread in a relative sense. I will give you another example. Let us talk about population ok. Let us say a country has a population of 1 billion consider another country having a population of 1000. There are countries like that. Now if you consider the spread let us say a population of that country over some years is your random variable. The spread in the first one having a population of 1 billion is let us say 1000 that is the standard deviation. Now you consider the other case where the population mean is 1000 and you have a standard deviation of 1000 same standard deviation. Would you call that the spread is same for the two populations? Not really right. So, standard deviation is not really a good measure of spread all the time. Did I get my point clearly to you? I will repeat that example again. You have two populations one having a mean of 1 billion and a standard deviation of 1000 ok. Another having a population of 1000 and let us say another having a population of 10000 and a standard deviation of 1000 ok. So, in which you see more changes of course the one with a population 10000 right because changes can be more drastic in this. Once you will have a population of 5000, once you will have a population of 150000, 120000 and so on. While the population close to a billion will not change that much if the standard deviation is only 1000. So, it makes sense to divide that standard deviation by the mean to have the idea of equivalent spread ok. Now if you compare these two statistics in terms of the coefficient of variation in one case you have 1000 over 10000 ok 10 percent coefficient of variation. In the other case you have 1000 over a billion much much less ok. And it makes sense to compare these two in terms of the coefficient of variation rather than in terms of the standard deviation, but you cannot apply this thing all the time particularly when the mean is close to 0. If the mean is close to 0 then the coefficient of variation goes towards infinity and mean having close to 0 is not a big deal. Many a times you are dealing with random numbers which go towards the negative direction and positive direction both and you might have a mean close to 0 ok. Ideal value close to 0, expected value close to 0. There it makes no sense to divide the standard deviation by the mean and compare the coefficient of variation ok. So, you have to use a judgment to see which is a better measure of dispersion, standard deviation or coefficient of variation ok. There is another related parameter covariance which you define for two random variables when you have a joint distribution of x and y. Then instead of variance we use the term covariance which defines a joint spread between the two ok. And how do we define it? The definition you can see is very similar to the definition of a variance for a single random variable. For a single random variable x what we have is expected value of x minus mu x whole square. Now, since we have another random variable we keep the dimension same what we do? We replace 1 x minus mu x by y minus mu y. So, we have the measure of joint dispersion in a joint distribution covariance written like this covariance of x y is measured at the expected value of x minus mu x times y minus mu y ok. We will not go into the detail of this only when we go into again continuous distributions probably this will be more useful. Now, for the later part of the talk we are going to discuss about some special discrete random variables. There are various common ones which we use and apply in our daily calculations in science and engineering. The one most common is the Bernoulli and binomial distributions. When you have the Poisson distribution hyper geometric, geometric, negative binomial and so on. We will discuss the first two start with the Bernoulli and binomial. So, x will be a Bernoulli random variable if for an experiment the outcome of the trial is binary. That means you have either this or that no other only two options. So, it could be failure and success and you denote 1 by x equal to 1 the other by x equal to 0. That is the normal way of denoting a Bernoulli random variable or the result is yes or no ok. Does it occur or does it does it not occur that is what yes or no. You put a value equal to 1 for this you put a value equal to 0 for the other that is a binary outcome. For that if you can write that probability of x equal to 1 is small p automatically probability of x equal to 0 becomes 1 minus p because there are only two outcomes. If that is the case then x is a Bernoulli random variable. What is the expected value? We compute using our basic formula. It is a discrete random variable. So, expected value summation of all x i times the probability mass function. So, there are two values 1 and 0, 1 times the mass function which is probability at x equal to 1 and probability of x equal to 0. So, this is 1 times p and 0 times 1 minus p finally, you get p. So, this is the expected value. Does that make sense? Now the outcome is either heads or tails you keep on doing the that. Then the expected value if the probability of head for each outcome is p the expected value over several will be equal to p that is a Bernoulli random variable. Then you go to binomial which is an extension from the Bernoulli random variable is that the random variable y will be a binomial if y is the number of successes in an independent binary trials. Both are very important. Independent is very important that two trials should not have any relation to each other. They are all independent. If this has occurred it does not mean anything to what is going to happen next. So, for n independent binary trials the number of successes is y and y will be a binomial if for each trial the probability of success is p and automatically the probability of failure is 1 minus p. So, it is an extension from the Bernoulli. What did you say about Bernoulli is x will be a Bernoulli if for each trial it is a binary outcome 1 or 0. Now you are saying y is the number of ones or number of successes among n independent trials independent binary trials. Then y is a binomial random variable. We will see example and see how and y is denoted as a binomial with parameters n and p that is how we write it. So, now we are looking at the mass function of the binomial random variable. So, mass function as always with a small p suffix capital Y at y is n choose y p raise to y and 1 minus p raise to n minus y. I want you to spend some time on this and see if this is reasonable or not. Look at the two quantities p raise to y and 1 minus p raise to n minus y. So, we say what is the number of success y and the number of failure automatically is n minus y. Probability of success in each trial is p. So, p raise to y based on the basic principle of counting if you get back to two lectures earlier and probability of failure is 1 minus p multiplied so many times that means n minus y times number of failures. How many ways you can choose number of successes y out of n that is n choose y. So, you have this formula of the mass function that means y equal to y number of successes in n independent trials equal to y is n choose y times p raise to y times 1 minus p raise to n minus y. Of course, for y equal to 0 to n only whole numbers. So, binomial random variable of course, only whole numbers you cannot have anything else. So, many successes out of n has to be whole numbers and has to be between 0 to n can be negative can be anything else. And we can compute the c d f from this mass function c d f is how many times y is less than equal to y capital y the random variable less than equal to y number of successes less than equal to y. We can obtain that by summing this mass function from all values of y starting from 0 to small y that is what we are doing 0 to small y we are adding up this p m f values and we get the c d f. So, this formula gives you the probability that the number of successes is less than equal to y out of n independent binary trials. There are a lot of application of this it is a very common one if you look at any text book you will see a lot. Let us first look at the expected value of the binomial random variable and also the variance. And for this we will use a simple technique we will write y the binomial random variable as summation of x i i ranging from 1 to n where x i is 1 if the success of each independent trial is it is a success in each independent trial and 0 if it is a failure. So, is that making sense to you. So, let us say total you have 10 trials in 5 you have success. So, what is y 5 of course, and x i for the successful ones are 1 each. So, 5 times 1 you get 5 is that ok I will just go for a simple example show. So, and x i is 1 for success 0 for a failure. So, if we have let us say 10 independent trials. So, I will just write them down first one failure second one failure success failure success these are different x i values what is y summation of all this. So, we can write y as summation of x i is if x i denotes 1 for success 0 for a failure then y is summation of x i which is total number of success that is how we represent it. So, if we represent it that way then expected value for each x i is what we know p that is the probability of success and the probability of failure for each binary trial is 1 minus p. And then we can obtain the expected value of y as the summation of expected values of x i i going from 1 to n. So, the expected value of the binomial random variable y is n times p a simple way of looking at it. Similarly, we can also obtain the variance of the binomial random variable y. And again we use that same x i where y is the summation of x i x i denoting 1 for a success in an independent trial and 0 for a failure. And first you write that expected value of x i square is same as expected value of x i equal to p how is that let us go through that. So, expected value of x i square what are the two different values that x i can have 1 and 0. So, 1 square times its probability which is p plus 0 square times its probability which is p plus 0 square times its probability. So, finally, you have p which is same as expected value of x i. So, with that if we have e x i square equal to e x i equal to p then variance can be easily written as e x i square that is what we needed minus mu square mu is e x i which is p square. So, p minus p square is the variance of x i and from that you obtain the variance of y as the summation of variance of x i. You remember that formula that we discussed earlier I will go back yes variance of y which is a linear combination of different random variables is a i square variance of x i. And now for this y is summation of x i there are no a values. So, variance of y is summation of variances of x i values. So, n times this which is p times 1 minus p simple ways of calculating it. And now we go for an example of the binomial random variable. I cite one example again from a civil engineering background. So, what we have is for a construction company the record says that 70 percent of the projects complete on schedule. And the company wants to find out what is the average number of projects to be completed on schedule for the next 15 projects. This is a problem of a binomial random variable where the success is completion on schedule and failure is not completing on schedule. Now, we treat it as a binomial random variable how? We consider completion of each project to be independent. We said that for a binomial random variable y for n independent trials remember that independent word was very important. So, this is the first assumption that completion of each project is independent may not be always true in the real life, but let us say here that is what it is. So, x is a binomial with parameters n p where n is the total number of independent trials 15 and p is the probability of success that means completing on schedule which is 70 percent 0.7. And we want to find out the expected value. We said what is the average number of projects to be completed? What is the binomial random variable is the number of success over n independent trials? We have 15 independent trials what is the number of success that is the random variable x and we are trying to find out what is the average value expected value of x. So, the mass function we know the formula n choose x any random number x. So, 15 choose x times p raise to x 1 minus p raise to n minus x and if we have the mass function from that we can easily obtain the expected value. How exit value is x times the mass functions summed up over all the trials that means 1 to 15. So, you have this formula this should give you the average of the expected number of projects completed on schedule out of the next 15. So, you can compute that and get the result. Alternatively you can also use the formula that I gave just two slides back that expected value of the binomial random variable x is nothing, but n p where n is the total number of trials and p is the success in each independent binomial trial. So, 15 times 0.7 equal to 10.5. Now, what you can do you can check yourself is if this formula gives you 10.5 or not. So, similar concept you can apply for various examples you know completion or non completion occurrence or non occurrence for all independent trials. So, you can apply it for having a flood of this intensity occurring a year that is on independent trial. Non occurrence in a year will be the failure. So, occurrence in a year is probability p, non occurrence in a year is 1 minus p. Now, you can compute what is the likelihood of having such floods in next 50 years. That will be a binomial random variable and you can compute how many times you expected to occur in next 50 years. Another common discrete random variable is the Poisson random variable and you see that we are writing p in a capital letter because that is based on a name of the person who first defined it. So, x will be a Poisson random variable if x can take values like 0, 1, 2 and so on only whole and positive numbers starting from 0 and has the following mass function and you have a formula like this e raised to minus lambda or lambda raised to x divided by factorial of x where lambda is greater than 0 it can be it can be a fraction it can be any other number, but greater than 0. Then we call x to be a Poisson random variable with parameter lambda. Here you just have a formula I will explain how to use it. The fun thing about this random variable is that it has the same expected value and variance both equal to lambda. Does it mean it has a very high spread same expected value and variance? No spread is measured by variance but it is the square root of variance that is the standard deviation which we usually use for that. So, depends on what is the value of lambda that will tell you what is the spread. You can check any book like the textbook that is being referred all the time process book on introduction of probability and statistics to engineers and scientists I think that is the name anyway. So, you can check any book for the proof that both expected value and variance is equal to lambda. Now application a Poisson random variable can be used to model the possible occurrence of an event at any point in time or space. So, given an interval could be in time could be in space what if this event occurs here or there or how many times in this range that can be very successfully modeled using a Poisson process. So, this is what a Poisson process is very model occurrence of things like earthquake over time. What is the likelihood that this earthquake occurs over next 20 years or earthquake in a seismically active location? Given this region take the San Francisco Bay area what is the likelihood that it occurs here or there or there in space you can also distribute it in space or the other example for distributing in space is the occurrence of fatigue cracks at some location in a well. You know what welding is right where you correct two metals using welding and they are very likely to have fatigue kind of cracks fatigue means which you have for repeated loading and unloading. So, if you want to locate for this length of weld what is the likelihood of having it here, here, here and so on that kind of process can be modeled using a Poisson distribution. It is very important to use this term mean occurrence rate in a Poisson distribution which is the average number of occurrences per unit time for a given interval t. Time can be replaced by space also. So, mean occurrence rate is average occurrence per unit for a given interval and it is denoted by nu and nu is equal to lambda that parameter which defines the Poisson distribution divided by t is the interval even in space. And Poisson become close to binomial if lambda equal to n p and n is very large and p is very small. I will try to give you some more example later how it becomes like that. Let us go for an example. We have this information that there have been three large earthquakes large meaning magnitude having more than 6 degree magnitude. So, we had three large magnitude earthquakes in the last 50 years in a given region let us say Uttarakhand and this occurrence can be modeled as a Poisson process. Then the question is what is the probability of having any such earthquake in the next 10 years? Very common use of a Poisson distribution. So, x is a Poisson process with nu the mean occurrence rate equal to 3 that is divided by 50. So, it is 0.06. So, with t equal to 10 and lambda equal to nu t we have what is the probability that you are trying to find out that the earthquake happens. That means, it happens at least once the probability x greater than equal to 1 is of course, probability of 1 minus not having it at all 1 minus probability x equal to 0. So, you put it 1 probability x equal to 0 is the mass function at 0. So, we use the formula again I will show you the formula it was e raise to minus lambda lambda raise to x divided by x factorial x is 0. So, you put that. So, this is the probability 0.451 or 45.1 percent that there will be at least one earthquake of magnitude 6 and larger in the next 10 years. Is that ok? Did you understand this? Similarly, you can also find what is the probability of having no such earthquake in the next 10 years and that will be no such earthquake means x equal to 0. So, p m f again at 0 and this is the one again I mean that would be definitely 1 minus the probability that you found earlier. So, it is 54.9 percent. So, these are occurrences of Poisson distribution. There are many other look at any book you will see many other examples things like discharge of alpha particles in a radioactive material and so on. You can model this using a Poisson random variable that concludes today's lecture. So, looking back these what we have gone through. Expected values, variances, their properties, binomial distribution properties as example and Poisson distribution properties application example. Thanks.


Alright guys, welcome back to the channel. If you're new, man, it's Bobby. Interesting video today, we're gonna check out how can Jesus be both God and human on the channel Questioning Christianity. Despite the title and the channel's name, the producer of the video is actually a Christian himself. So today we're gonna look into the psyche of a Christian, so to speak, and how he justifies the statement that Jesus is supposedly God and man at the same time. Guys, before we jump into the video, if you enjoy my content, leave me a thumbs up, subscribe to the channel if you haven't already, and check out the links in the description box below to further support. And now, with no further ado, let's have a look. How could Jesus not know something like the day or hour of his return, if as God, he's meant to be or not? If the second person did the Trinity upholds all creation, how did the universe not cease to exist or else fall into chaos when Jesus died on Calvary's cross? And how exactly does the infinite become finite? Well, these and other questions don't simply come thick and far from Christianity's critics, but they're also pondered upon by curious Christians, who like me are awed by the beauty of Christ's incarnation, this bridge between heaven and earth, but who also may be struggled to comprehend how it all works, and especially how to capture it in words. So how do the reasons of the Christian story weigh in on the mystery and beauty of the word becoming flesh? How can Jesus be both God and human? For starters, if you wonder why historic Christianity holds that Jesus is God incarnate, or God the Eternal Son becomes human, check out our other video, Is Jesus God? There we lay out the case from these dozens of places in the New Testament, where Jesus and his apostles clearly believed and taught that Jesus is God. From his unparalleled power of creation and authority to forgive sins directly, to his personal claim to divine titles and his welcome of worship, right down to his resurrection from the dead as the prophesied sign vindicating his claims, the positive evidence of Jesus' divinity is everywhere. So let me share a few things that I found helpful in wrestling through the logic of the Atonement, how it answers Christianity's critics, and what it means for you and for me personally if it is true. First, a scripture draws helpful boundary markers around orthodoxy. Even though all the raw materials of Jesus' humanity and divinity are there from the first chapters of the New Testament, it took time for the apostles and early generations of Christians to construct their proper theological edifice to give them full expression. Enter what's known as the... It doesn't make any sense. He has expected a lot of word games, a lot of mental gymnastics, but ultimately what he has to admit is of course that the Trinity was not preached by Jesus nor his direct apostles. There is no mention of the Trinity and my argument is very, very simple. I say if Jesus is God, God came to earth. Of course he would come with a message that is crystal clear and he would point out the differences to the previous messages because up until the moment that Jesus incarnates allegedly, you had many, many prophets that preached pure monotheism. They were preaching to the idolaters and they are telling them to not worship the golden car for example, but to worship God alone. But now Jesus quote unquote, God comes into this world and has to teach you about the Trinity supposedly, but there is no mention of that in the Bible whatsoever. There are a lot of vague passages I admit and therefore 325 years later the church fathers at the council of Nicaea have to discuss and vote democratically if Jesus was the son of God, if he was God himself, if the nature of God is a Trinity and whatnot. There is so much confusion, so much dispute in the teachings of Jesus Christ or what remained of them that they needed 1800 priests and bishops and whatnot that had to discuss and vote if the Trinity is the right doctrine or not. That just shows you clearly that Jesus never preached it. While there were a number of councils and creeds that served these precursors, in 451 AD an ecumenical gathering of over 500 bishops hashed out the enduring theological boundary markers for orthodoxy. Essentially they sought to capture in words a base on the authority and affirmations of Scripture, what should Christians believe about Jesus's incarnation? Now I know this may seem like deep theological territory for any newcomers watching, but what they thought was the right thing. So it might appear as deep theological discourse and whatnot, but ultimately it's very very simple. You said it yourself. Over 400 years after the death of Jesus Christ, the proto-orthodox Christians still didn't know what their creed was. They didn't understand the nature of God. Hence we have to admit that Jesus Christ did not sufficiently apparently convey the message. Apparently things got lost and those people 400 years after Christ did not know what the true reality of God is and therefore they had to discuss, therefore they had to vote as I said 400 years after the death of Jesus Christ, they come to the conclusion, okay therefore it needs to be a trinity even though we certainly know that Jesus Christ did not preach it himself. What they landed on was the language of the hypostatic union of Christ, which is shorthand for the affirmation of three core beliefs. One, that there is only one person Jesus Christ. Two, that Jesus Christ had two natures, divine and human. Jesus was fully God and fully man. And three, that these two natures, while united and indivisible, are yet distinct and preserved. Not thrown together in a blender that either changes, confuses or corrupts their nature. This Chelsedonian Confession explains why it makes sense. The hypostatic union doesn't make sense because it presupposes that the second head of the Godhead, the second person of the Godhead of the trinity is the Son. The Son is fully man and fully God. But the same time there is the admission of course that the Son, the Word of God incarnated and became man. Therefore this nature had to become something. They of course claim that it has been eternally begotten and it never changed, but this doesn't make sense because prior to the incarnation the Word of God, the Son, was not man. And this is why they create another loophole yet again. Yeah, this is all through the hypostatic union. No, it is not. It tells you simply that there was the Word of God that incarnated into the flesh and now has a human experience. The Father on the other hand has no human experience. So experientially speaking, the Son is even further above than the Father. Don't you see? This of course circles back to the equality argument. How can they be co-equal if they have different experiences? This doesn't make any sense whatsoever and it doesn't befit God. God is absolutely free from need of experience because he is already perfect. Christological heresies had been soundly rejected. Running a fowl of the laneway mark is in either direction by denying the full divinity or humanity of Jesus. So that's what Christians should believe from Scripture. But translating their what of theological affirmation into the how or coming up with a workable model of the incarnation that can answer the opening salvo of skeptical and curious questions, that tends to be where Christians disagree. Which is my second point. Christians have options for modeling Christ's incarnation. Over the centuries, Christian theologians and philosophers have offered a range of models for making sense of the data of Scripture. The Canotic models from the Greek Canosis, meaning to empty, they interpret the Christ, him and Philippians 2 as saying that the eternal Son emptied himself of some divine attributes in order to become fully human. That's the argument that he is sometimes fully man and sometimes fully God, but at the same time he is always fully man and fully God, right? But when he says about the last hour only the father knows, well this is when he is fully human and shows you his full human side, just swapping back and forth. This doesn't make any sense whatsoever. As a more recent set of models, historically these theologians argue that God the Son willingly gave up or set aside whatever omnis were not compatible with being human. That means that Jesus was not omnipotent, omniscient or omnipresent. Even though by identity or relationship they argue, he still present. Yeah man, because it goes directly against the claim that he is fully man and fully God.  So silly. So how do these Canotic models explain the travel pastors like in Matthew 24 36 where Jesus says he does not know the day or hour of his return? Well easy, these Canotic theologians simply say Jesus was ignorant of the emails because God the Son had given up the attribute of being all-knowing to become Incarnate in Christ. At least setting it aside. Then there are the cryptic models from the Greek cryptos meaning to conceal which interpret the Christ him in Philippians 2 not as saying the eternal Son gave up divine attributes but that he veiled his divinity when he added humanity to himself. Not grasping onto but rather full. There you go, it's a contradiction after contradiction because there he just admitted after adding humanity to himself. This yet again presupposes that the word of God was not human prior and had to go through a human experience which in turn means that God needs an experience which goes against God because God is already perfect and is free of need. Alhamdulillah for Islam, it's so insane. Comprised for a number of specific proposals that stand in various traditions, these cryptic models say that Jesus still possessed the omnis but that he could choose when and how to give the expression in order to fulfill his redemptive purposes. Almost like the undercover box. So how do the cryptic models then answer the travel passage? Well that depends on which version of these models. John Kelvin for instance made sense of Christ's ignorance in Matthew 24 36 by saying that despite Christ's divine mind being omniscient, according to his limited human mind, Jesus did not know the day or the hour. This doesn't make sense. I mean at this point I'm just repeating myself but just for good measure, yet again if he doesn't know something that the Father knows, that then implies that he is not fully God.  Jesus did not know the day or the hour. For Kelvin and other cryptics, the distinct two natures of Christ in the hypostatic union, the finite and the infinite, opened the door to saying ignorance and omniscience can be true of the one person Jesus at the same time, depending on what nature is behind the stars. It couldn't be that he is a messenger of God. And that the divine loveless retains all his attributes and is united to yet not entirely encompassed within the human nature of Jesus? That means Christ's death on the cross, it does nothing to reduce the trinity to a binity or to cause creation to descend into chaos. But this cryptic proposal of two minds, it's unsatisfying to some current theologians and philosophers like William Lane Craig and J.P. Moreland, for instance, who proposed a controversial third option, turned the Neo-Apollinarian model. On this model the Divine Logos, God the Son, supplied the human body of Jesus with all the attributes necessary for a limited human consciousness that had to develop and learn across time. But with all of the superhuman elements of the Divine Logos being located in the subconscious layer of Jesus. Yeah, all I'm hearing is partially this, partially that. That's it. It's a demigod, basically. Like in Greek mythology. Yeah. And as with the other models, this one can easily account for the various scenes in the Gospels which seem to show Jesus as limited in knowledge or power or as being subservient to the Father's will. Yes, but not as fully God. Who knows? The conodic model potentially runs afoul of explaining how the omniscient Yeah, all of this brings us back to the meme. You ask 10 Christians about the Trinity and you get 10 opinions. Alhamdulillah for Islam. La ilaha illallah. There is no God worth your worship, but Allah. Everybody agrees. No matter if you look into different sects or madhaps or whatnot, the core of the religion is always the same. The absolute unity and transcendence of God alone. Or how Jesus can be fully God without any claim to these defining attributes. Yeah, man. The Neanderthalian model potentially runs afoul in the other direction, as some charge it as a divine possession model, where Jesus has a less... Your Lord is not a Lord of confusion. And the purpose of this model is to reduce differences in two minds with stunted explanations of how they interact in one person. Now, if you're confused, that's totally okay. Yeah, it's totally okay. The purpose of this whole survey in philosophical terms. As I just said in the Bible, it says that your Lord is not a Lord of confusion, but if you are now confused, the creation says that it's totally okay. Just relax and accept it. Our doctrine is confusing. Let's all accept Christianity today. If you're confused, that's totally okay. The point of this small survey in philosophical options is to show that each model can go a long way towards answering the speeders of critics and the curious questions of Christians. And if it is possible for us as finite and fallen human beings to come up with possible models to make sense of God's revelation, then shortly Christianity is off the hook. For an infinite and all-wise God has vastly superior explanatory resources than anything we can construct. No, Christianity is not off the hook whatsoever, because the claim is that previously to Jesus Christ we had prophets. All of those prophets came with the same message. Worship one God alone, do not worship idols. And now finally, the grand finale, God himself incarnates. And he comes with a new message. This is what you claim, but to this very day you're still confused and don't know what that message is or what that message was. And therefore you need to reinterpretation from the truth that is allegedly guided by the Holy Spirit. But even those people that are guided by the Holy Spirit still have no answer to the Trinity. Could this be a false doctrine? So we don't need to get too caught up on the exact mechanism. No, don't. Like C.S. Lewis said of atonement theories in mere Christianity, when it comes to our models of the Incarnation, the thing itself is infinitely more important than any explanations that theologians can bring. Okay, why? Which leads to my third point. Why? Christ's Incarnation transcends mere explanation. The Christmas story of God, the Eternal Son entering our human story, is told through supernatural signs and cosmological conspiracies. And the whole thing, it's loaded with mystery and meaning. A peasant parent's pluck from obscurity, Yeah, sure. It's packed with mystery and meaning, and this is why we find very, very similar stories within Greek mythology, within Egyptian mythology, and many other pagan mystery schools from back in the day. The rising after three days, the defeat of death, those stories have been known before. Denisius, for example, is another character that is extremely similar to the Christian perspective of Jesus Christ. So therefore mythologically speaking, yes, those stories have value, they have archetypes that can teach you something, but that doesn't make them true. Now, if you're wondering what difference this all makes, think of the Incarnation as God's response to life's deepest question. Things like, who am I? What is my life's purpose? Is God even real? How does he feel about the stuff that he bought in the world? Has God really a man? Tell me. Can I change? Is there hope? Now, in answer to these questions, God could have sent a why, but he doesn't. Why? Well, because words come cheap. It's the biggest why. What sends us a who? It's the biggest how. He comes himself as Jesus, being God's answer wrapped up in a human being.  But do you? As human, Jesus experiences our life so that he gets us, feeling every pain and loss and temptation and betrayal common to our raw experience. That's not true either. Man, this guy really likes to talk, but it's all falsehood. Jesus Christ's experience, Jesus Christ's human experience is nothing like ours. He didn't get married. He didn't have children. He didn't have the same hardships like ours. If we suppose that he got crucified, of course, that is a huge burden that he had to go through. But nevertheless, his life is nothing like ours. At the center of God's purposes in the incarnation, as both God and human, Jesus is uniquely qualified to heal the rift between the two as an all-sufficient savior. A paying a price that only God could pay, but also one that only humanity should pay. This is but a glimpse of the good news wrapped up in the word becoming flesh to dwell among us. So please, you're welcome to wrestle with the logic of the incarnation. Truth invites questions. There is no logic. And so Christians have sent out a range of reasonable options in trying to make sense of God's revelation. But the beauty of our Creator entering his creation, it transcends mere explanation, which is why more than any sophisticated models of the incarnation, I've always found that the Christmas story is best experienced and expressed through songs. Songs that capture the mystery of the incarnation and songs that back in all who hear them I'll just sing a song guys. That will explain it. Don't fix it. The flesh that God had seen. Hail the incarnate deity. A pleased as man with man to dwell. Jesus our Emmanuel. All right, that's it for today's video. I'm going to cut it off here because I said pretty much everything I needed to say throughout the video. It all boils down to, hey guys, there's a mystery. Hey guys, it explains how, why night actually doesn't. You are maybe confused, but that doesn't matter either. Ultimately so beautiful. Listen to songs guys. Bro, then I can become Hindu as well. Those guys love to chant songs and they believe that God incarnated in millions of Abba tales. How? I suppose it's a mystery after all as well. Why not go with that doctrine? As if life is not confusing enough, let's make it even more confusing with confusing mythology. Yeah, that makes sense. Why would we use our God given intellect and actually look at the scripture ourselves and recognize the red thread, the red thread of prophets being sent with one message. Worship God alone. Worship the Creator. Do not worship the creation. Do not worship sticks, stones, the golden calf, other humans, pharaohs and what not. Simply worship the invisible God. Worship the ever-living sustainer of the universe. Worship him alone. This is what we can find within the scriptures. Why would it be any different for Jesus? Especially if we take into account that his message has not been preserved. Apparently you can see it yourself because even the highest church fathers cannot make any sense of it. 300 years after Jesus Christ's demise, they come up with the Trinity. They have to reinterpret yet again 600 years, 700 years after his death. They still do not know if they should venerate icons or not. The original message of Jesus is obviously lost. But if we look into the scripture ourselves and we use our God given intellect, our God given rationale, of course we come to the conclusion that there is only one Creator that created everything. He is the only one that we should worship and this is why prophets have been sent over and over and over again. But anyways, ultimately this is a test of free will. If you want to stay within your confusion and you want to worship a man, a God, maybe both through hypostatic union, I don't know, maybe I'm going to sing a song yet again for Christmas. That will make sense. I'm wrestling with my faith. It all doesn't matter because it is all so beautiful. Okay, you do you, I do me. Alhamdulillah for Islam. All right guys, but this is it for today's video. If you liked it, leave it a thumbs up, subscribe to the channel if you haven't already and check out the links in the description box if you want to further support my work. And now, as always, make God one God. Bless you all, much love and peace.


Jesus is going to where the need is. And this whole thing, building a radiant city, is about us mobilizing to go to where the need is.  All right, I want you to turn with me to your Bibles today to Matthew chapter 28. Matthew chapter 28, this is part five. It's the final part of our Be Radiant series. What we've been doing over the last five weeks is looking at our core values, the pillars of what it means to be radiant. When we say be radiant, we're talking about a genetic code of what we believe Jesus intends for every believer, no matter what church they go to, no matter where they live, he has a certain profile that he wants his followers to live like. And so these aren't just unique to radiant church, although we hope that we exemplify those.  It's living our life-centered anchor to the word of God. Number two, it's being spirit-empowered. The Holy Spirit continually filling us and empowering us so that we can fulfill the Jesus mission. Number three, it's being family-oriented. That we love and we relate to God as a father and we relate to one another as family. That's what the church is meant to be. Number four, we talked about this last weekend, which is kingdom of God focused. That we need to have our eyes set on Jesus so that everything else comes into perspective, so that the prescription that our heart, the lens that our heart is seeing through is not of this world, but it's of the kingdom of God. And number four, what we're gonna talk about this weekend is mission-motivated. We're called to be mission-motivated. And this is one that my heart beats 24 hours, seven days a week connected to this particular value because I believe all of the other ones lead to this one. Look with me here at verse number 18. We're gonna begin in verse number 18. And Jesus said to them, He came and He said to them, all authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father, in the name of the Son, and in the name of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always to the end of the age. This is Jesus' last conversation with His disciples. It's on the heels of His victorious resurrection from the dead. He's about to ascend to the throne and to reign and rule from heaven. He promises to return to establish His kingdom on the earth at the end of the age, but He commissions the church, His followers to go into all the world because He says all authority in heaven and earth has been given unto me. What authority is He talking about? He's talking about the authority that Adam lost in the garden that gave the enemy access to bring darkness into the world and to hold humans hostage bound by sin and to corrupt every aspect of God's good creation. Jesus won that back at the cross and He was vindicated by His resurrection and now He says all authority belongs to me. And because it belongs to me, He says to His disciples, I want you to go into all the world and I want you to make disciples of all nations.  This is called the Great Commission. I say it's the only commission because Jesus didn't give us one really important commission and then give us a whole bunch of other things that we could do that were optional. No, the only reason Jesus sent the church, built the church, established the church, gave His Holy Spirit to the church was so that we would do this one thing that we would go. It's a simple two letter word, go. Do you know that two thirds of God's name is go? Do you know that if you look at it backwards, two thirds of God's name is do? That'll get ya. But it's important for us to know that this commission was not just given to His disciples 2,000 years ago. You and I are the extension of this nucleus, critical mass, small group of disciples and the last instructions from our Lord, King, and Master was to go make disciples, preach the gospel, baptize believers and to teach them everything that Jesus taught and to wait for His return. This is our job. This is our calling. This is our commission from heaven and it's not just a calling for a few people. It's a calling for every single disciple of Jesus. But unfortunately, in the church, over the course of time, there are moments when the church loses its focus. It's not just here, it's not just now, but it happens over the course of 2,000 years. You can see the cycle repeat itself over and over and over again where the church experiences renewal, revival, the Holy Spirit is poured out and there's passion for Jesus and because we love Jesus, we want Him to receive everything that He deserves. And then we have missions, movements that thrust out into the world but then eventually it begins to cool down, the embers begin to grow cold, we get focused on self, we get comfortable and we relegate the great commission to books that we read and really radical believers that we're just glad that they're willing to go on our behalf. And instead of it being the great commission, it becomes the great omission where it's the one thing that we should be doing but it's the only thing that we're not talking about. We're supposed to be people that are motivated, I mean motivated for the mission, not tolerating the mission, not applauding the mission so somebody else goes to it but this is supposed to be our heartbeat. This is supposed to be what we're passionate about. Oswald J. Smith, who's a great pastor, a great leader, he said this, the mission of the church is missions. This is our mission, it is missions, this is it. And maybe something that you did not know about Jane and myself is that before we moved to Kalamazoo to plant, radiant, when we were first married in 1992 and Jane quickly thereafter became pregnant with our firstborn Ashley, we contemplated, in fact, we were fully funded or pretty close to fully funded to go on the mission field. Jane and I were gonna move to the Philippines and we were gonna work with a church planting organization in a Bible college in Manila that was planting churches across thousands plus islands in the Philippines. We had met with the missionary organization, we had met with the leader over there, our church that we were a part of had decided that they were going to send us, they were gonna give us finances, we had other people that were giving us finances, we had told both sets of our parents, made her parents cry, my parents were like, wow, where did that come from? And I'm just like, because Jesus called us to go.  Jane's like, how am I gonna have, I said, you're gonna have the baby in the Philippines. And she's like, I don't wanna have a baby in the Philippines. I said, it's for the cause of Jesus. And we were starting to make plans to move on to the mission field. We were gonna make our life about the Great Commission. And on a summer day when we were sitting on her mom and dad's little lake house down by the beach, I was sitting in a lawn chair and I was reading out of a book by a man named Watchman Knee, a book called Concerning Our Missions. And I was reading, it was page 47, left side of the book on the bottom paragraph, page 47, third sentence in. I still remember it, it's blazed on my imagination. And he said these words that no amount of ambition, no affirmation from leaders or men's well intentions can substitute for the call of God. And when I read that, the Holy Spirit whispered to me and he said, I'm so glad you're willing to go, but I have something else for you. You will do more for global missions if you will plant a missions motivated church than if you will go yourself. And it was because of that moment I turned to Jane, literally I turned to Jane and I said, I don't think we're supposed to go. And she said, okay. And I know she was bummed. And that was the thing that set into motion, us deciding we're gonna plant a church. And I'm gonna tell you something. When we planted this church, some things that maybe if you have come along over the last 20 plus years, maybe you don't know, but number one, the first thing that we said to the Lord when we planted this church was God. Number one, if we can't be a blessing to the nations of the world, then we will go and do something else because there are enough great churches, but we wanna be a mission sending, mission motivated church. And so the very first check that Radiant Church ever wrote, check 1001 was to our missionary in Latin America. Before we rented a building, before we paid for anything, we did that intentionally. We wanted to make a statement that Radiant Church was going to be a missions motivated church. Another thing you may not know, but over the course of probably the first 10 years of our church, every Sunday when we would gather at the end of worship, one of the things that we did was we prayed over unreached people groups or we prayed over a nation in the 1040 window or over one of our missionaries. We would put up on the screen a picture of either the people group or the nation or one of our missionaries in the 1040 window. If you don't know what a 1040 window is, it's 10 degrees long, two 40 degrees latitude. It cuts across equatorial eastern hemisphere of North Africa, the Middle East, Asia, and it represents over 85% of the unreached people group in the world. And so we did that for years. In the old sanctuary, which is now our student ministry's center at the Richland campus, when you would come in, we had I think 65 flags around the room that represented nations that we supported missionaries in. And we did that because we wanted to make a statement. In fact, in preparation for this, I started thinking to myself, I think we're gonna start praying for nations again. I think I'm even considering putting flags back up. I just think we ought to be able to see that we're a church that is impacting the nations of the world. We're not just a little building in the middle of a cornfield or imported or even an online presence, but we are a missions motivated church. And I tell you that because listen, we're 25 years in this year and I am more motivated for the Great Commission to reach every person on the face of the earth than I ever have been before. My fire for mission has not grown cold. And the motivation of our church, the fuel of our church is going to continue to be this because this is Jesus's last words to us. And here's what I know.  How many parents know what I'm talking about? Do not stay up past 10 p.m.   What time did you go to bed?  When Jesus returns, he's going to pick up right where he left off. He's going to look at us individually and he's going to look at the church of Jesus Christ and he's gonna say, what did you do with my commission? What did you do to preach the gospel? What did you do to make disciples? What did you do to bring the gospel to those who've never heard it? What did you personally do? What did you do with the resources that I put into your life? What did you use with the giftings that I put into you? Did you have a heart for the lost here and around the world or were you focused just on yourself? This is where Jesus is gonna pick up. And God has called us to be missions motivated. Today, what I wanna do is I wanna describe to you what being mission motivated looks like. Mission motivated people believe some things. So here's five things that mission motivated people believe. And here's my heart. Not that we would say check, check, check, check, but we would actually recognize that in all of our lives, if you're truly a follower of Jesus, here's what I know is on each one of these five things I'm gonna tell you, there's a slide scale of intensity. There's some of these that you're gonna be like, yes, 100% and some it's like, well, I'm more of a two out of 10. This isn't to judge or to condemn. This is for us to have a dashboard to then ask the Holy Spirit, Holy Spirit. I wanna be a 10 on all of these. I wanna burn with passion for the things that you burn with passion for. So here we go. Mission motivated people believe that number one, everyone is a missionary. Mission motivated people believe that every single believer is a missionary. Do you know who the first missionary was? It was Jesus. Jesus is the first missionary. What does John 3.16 say?  So Jesus came in the flesh. Listen to what Jesus said about himself in John 5.30. He said, I can do nothing on my own. As I hear, I judge and my judgment is just because I seek not my own will, but the will of him who sent me. Jesus was the first missionary because he came from heaven to our world, our culture, our existence in order to save us. He set the example. And then he sent the disciples and he has sent us every single one of us. And this is an attitude. This is a posture that every single one of us need to get a grip on, that we are all missionaries. It's not just a couple of people. I grew up in missionaries would come to my grandparents' small church and they would have like a slide projector, set it up and they'd show you pictures and here's a straw hut and here's a lion and here's our well and here's our little chapel or I remember as a kid thinking, I don't wanna get saved because if I get saved, I know Jesus is going to call me to be a missionary. I was scared that I was gonna have to wear a loincloth, learn how to throw a spear and I was gonna live in a jungle somewhere. I was just freaked out. I saw the elephants, I'm like, I'm out. And there are a lot of people who are like, well, missions is what takes place on the other side of the world and missionaries are the people who come back, once twice a year and they're the brave, they're the kind of the extreme Navy SEAL Christians who sacrifice everything here to go there to the mission field. But I want you to understand a missionary is someone who is sent by Jesus to proclaim good news and because of that, every single believer is a missionary.  Every single believer is a missionary. Charles Spurgeon, the Prince of Preachers once said this, every Christian is either a missionary or an imposter because it's in the very nature of what it means to be a Christian is to care, it's to love and it's to say freely I've received therefore I freely give. Every believer is a missionary. You might think well, I live in America, this isn't a mission field, I'll wait for a couple points down the list and you're gonna realize very quickly that you are called to be a missionary. Number two, mission motivated people believe that everywhere is a mission field. Everywhere is a mission field. Mark 16, verse 15, it's a different version, this is Mark's version of the Great Commission and Jesus or he said to them, go into all the world and proclaim the gospel to the whole creation. Let me give you some statistics here, around the world right now in the Middle East, Asia, North Africa and India, that represent the largest area of unreached people who've never heard the gospel or have not had a clear proclamation of the gospel, there are 7,422 unique people groups who are right now alive on the planet who have not heard the gospel, right at this moment. I'm not talking about nations where it's been preached over and over and over again. I'm talking about unique people groups that have never heard the gospel. So the largest mission field by culture is the Middle East. Second largest culture that is per capita would be Asia. Third largest mission field in the world globally is India, which is a subcontinent. The fourth largest mission field by culture, per capita is North America. Because if we define mission field by the population of people who do not identify as Christ followers or have not yet necessarily heard a clear understandable presentation of the gospel of Jesus Christ, then North America, the United States, Canada, represent the fourth largest mission field in the world. If you see yourself as living in a Christian nation and sending people to the heathen, then you have a broken mindset and you don't have a mission motivation because you see this as home base. But if you recognize God when he created me, put me in North America as a Christ follower, all of a sudden you begin to recognize that America is not so Christian after all. Come on, somebody. Well, why doesn't God send some missionaries there? He has. And we're all in this room and we're all joining in right now. We're missionaries. Well, I don't want you. Jesus sent somebody. He's like, I did, I sent you. Well, I don't know how to be a missionary. It's time to learn. You can YouTube everything else. We can DIY everything else. We can memorize everything else. Well, I don't know how to really share the gospel. Maybe it's time we figure it out. But if I don't see myself as a missionary, then I don't have any responsibility. But if I recognize Jesus put me here on purpose, it's not like God's up in heaven going, wow, I really didn't know America was gonna go like this. It's kind of crazy and all these poor Christians living, let's get them all out of there and put them in a little Christian commune out in Montana so that they don't get rattled too much and then we'll send some real missionaries into America. I appreciate the gospel. That ain't happening. If you're looking at the world right now and saying the world's going crazy, America's losing its mind. It's always had its mind lost. It's always been in sin. It's always been depraved. Pressure's just making it rise to the surface. But wherever sin abounds, grace does much more abound. God has not given up on America. He's actually set up America and he sent you to win it. He sent church like a radiant church and other great churches all over America to win this generation. And he's not gonna send in five missionaries into America. He's already got hundreds of thousands of them. He's just gotta get the church to have a change in their identity of recognizing I am a missionary. Do you know that in the last 10 years, America has experienced a shift for the first time from its founding and here's what it is. We actually now receive more missionaries into our country than we send out. Asia is sending, Korea is sending, China is sending missionaries to America. Latin America is sending missionaries to America. Do you know that Brazil is 25% born-again spirit filled? Guatemala is almost 40% born-again spirit filled. So we've got some Latino fire that is coming into America and bringing the fuego and we need it. We need it. Praise the Lord. Mas fuego, mi padre. More fire, my father, bring it. All right. You know that God's not at all threatened by atheism, Hinduism, Confucianism, Buddhism, Islam, secularism, Marxism. He's not at all intimidated by those. None of those intimidate the spirit of the living God. He's called you and I to see ourselves as missionaries and everywhere as the mission field. Number three, mission motivated believe that every church is a mission base. Ah, we gotta stop thinking of church as a cruise ship and start thinking of it as a battleship because a cruise ship, if you got onto a Navy battleship and you expected that to be a cruise ship, you'd get on there and be like, hey, where's the buffet line? They'd be like, snap to attention and drop down and get me 50. It's like, what kind of cruise is this?  It's like, I'd rather wear my Tommy Bahama little, you know, loosey-goosey. I'm just gonna be sitting in the lawn chair. No, this ain't time to be sitting in the lawn chair. We're not playing shuffleboard.   This is your bunk, this is your uniform. Why? Because we're going to war. And if we come into church, then come, well, you know, I don't, I don't really like the worship. The worship's not my preference. Can I just tell you, the worship's not for you. We're not here to worship you. We're here to worship him. Well, the message, you know, I just have my preferences. Look, on a battleship, we come into alignment not because it's easy, but because the cause is just. And that's what we gotta get to that we see as the church, we're a missions base to our culture, to our city, to our context. This is what we're called to. John Piper, who is a great pastor at Minneapolis. He's written a lot of different books. Here's what he says. He says, missions is not the ultimate goal of the church. Worship is. Missions exist because worship doesn't. Worship, missions exists because worship does not. What do I mean by that? This means Jesus, the Lamb of God, who was slain upon the cross, deserves worship from every corner of creation. And the only reason he's not being worshiped is because the gospel hasn't penetrated that. And so we, as people who love the Lamb of God, we say, I'm going to the four corners of the world so that those people can worship Jesus worthy is the Lamb to receive glory and honor and power and glory and dominion forever and ever and ever. This is what we're in this for. It's not for our own glory. It's for his glory. Number four, we believe that everything must be fueled in prayer. Missions motivated people understand the connection between prayer and mission. Acts 4.29, the church is gathered. They're being persecuted. They don't know what to do. They're preaching the gospel. The local authorities have come to them and say they've thrown them in jail and then they've told them, if you keep preaching in that name, we're gonna throw you into jail. We're gonna throw away the key. And they said, well, is it right for us to obey God or obey man? We're gonna keep preaching the name of Jesus. And so they went and they started praying and here was part of their prayer. They said, and now Lord, look upon their threats and grant to your servant to continue to speak your word with all boldness. While you stretch out your hand to heal and signs and wonders are performed in the name of your holy servant, Jesus. And when they had prayed, the place where they were all gathered together was shaken and they were all filled with the Holy Spirit and continue to speak the word of God with boldness. You wanna know where boldness to preach and to be mission motivated comes from? It comes from the presence of God in prayer. In the early 1700s in Europe, there was a very wealthy aristocrat. His name was Count Nicholas von Zinsendorf. It's a great name. His father was very wealthy, was in politics. As he was coming up, stirring in his heart to study the Bible to know God was stirred and they had massive land holdings. He didn't wanna go into politics. He wanted to study theology. There was a group of believers, the Christians that were being persecuted in other parts of Europe named the Moravians. And he invited the Moravians to escape their persecution and to come and live on his property. And what is kind of the modern day area of Czechoslovakia, Germany, kind of in that area there. And he had a large estate and he invited them to come. So over the course of time, four to 600 Moravians moved there and they created a community called Hernhut. And in this community, they began to study the Bible. They began to pray, but they also experienced some differences, conflict. And Count Nicholas von Zinsendorf was used by God to kind of be a leader of this group. He pulled them all together one day in the mid-1720s and he said, we're gonna come together and we're gonna pray. And when they prayed, it was called the Moravian Pentecost. God moved in a powerful way in that prayer meeting. That prayer meeting ended up lasting 100 years, 24 hours, seven days a week for 100 years. God poured out his Holy Spirit where there was disunity, there became unity. And what was the overflow of that prayer meeting? Something powerful happened among that group of Moravians. They began to recognize in the place of prayer that Jesus, who is the Lamb of God, deserved the rewards, the full rewards of everything that he suffered on the cross. And what that meant was somebody had to go and to preach the gospel. So many of them became missionaries to different parts of the world. Some of them wanting to bring the gospel to the West Indies actually sold themselves into slavery so that they could preach the gospel to people groups on the West Indies islands that nobody had ever penetrated with. And so while they're on their ships, having sold themselves into indentured servanthood and they're waving goodbye to their friends knowing they're never gonna come back again, they shouted from the side of the ship, worthy is the Lamb to receive the rewards of his suffering. And it sparked a massive missions movement. You see missions motivated people understand that in the place of prayer, when we encounter the presence of the Lord, our heart is softened, not only to the reality of how beautiful, how glorious Jesus is, how worthy he is, but also of how much his heart is focused on the nations of the world. See, right now our heart's not necessarily broken for the nations of the world or even our own city or our own community, the way that God's heart is broken. When God looks at Southwest Michigan and he looks at even North America right now, his heart is broken for a generation that is having their faith deconstructed by TikTok theologians. When Jesus looks at America, his heart is broken because there's far too many churches where there is comfort and cotton candy preached and not the power of God that leads to salvation that's called the gospel. When Jesus looks at America, he doesn't see systems of education as high academic mausoleums, he sees them as concentration camps with barbed wire being wrapped around the minds of a generation, keeping them from the living God, the one who died for them, the one who loves them, the one who knows their name, the one who has their names carved in the palms of his hands. When Jesus looks at North America, when he looks at Kalamazoo, when he looks at the poor, when he looks at the broken, he hears the cry of the single mom who doesn't know how she's gonna take care of her kids. She hears the cry of the little child that's been neglected and left behind and is insecure and this last year has wreaked havoc on their soul.  Listen, we're not planting campuses so we can build an organization. We're not building prayer rooms so that we've got really cool buildings downtown. I could care if Jesus would let me, I'd drop the keys on the desk and I would go to the mission field tomorrow. I'm not doing this for my name. I'm doing this because Jesus has called us to care about the things he calls. He cares about. He's called us to build the kingdom of God one person at a time by picking them up off the ground, by filling them with hope, by making disciples of them, by baptizing them and washing off the filth of this age and seeing them come to life in the Holy Spirit of the age to come. He's called us all to be missionaries. Every single one of us. Every day I wake up and one of the things I do in my devotions is I remind myself, Lee, I am a missionary. I get the privilege of being a missionary to Kalamazoo, Michigan. I'm not a clergy member. I'm not a right reverend. I am a missionary. And you know what? There's nothing I'm prouder of than to be a missionary with all of you, to the city that he has called us to, to the generation that he has called us to because number five to be mission motivated means that we use every available means to preach the gospel. Spurgeon once said, if there is any one point in which the Christian church ought to keep its fervor at white hot, it is concerning missions. If there's anything about which we cannot tolerate lukewarmness, it is the matter of sending the gospel to a lost and dying world. A man named C.T. Stud once said this, if Jesus Christ be God and died for me, then no sacrifice can be too great for me to make for him. Church, we are called to be found at work in the harvest fields when Jesus comes. And I want you to think about not just mission fields as an ethereal kind of picture of continents and maps.   I want you to think about those faces. I want you to think about the faces of people that you passed today on your way to church. I want you to think about the faces of people that you work with, that you go to school with, that you follow on social media. I want you to see those faces and let the Holy Spirit pop them up because this is not about maps. This isn't about inanimate objects. This is about people. Lord Jesus, would you move our hearts today with what moves yours? Lord, this whole thing that we're embarking on called bidding a radiant city. It's not about a brand, somebody's logo, somebody's name on it. Lord, we want you to receive the rewards of your suffering. Lord, we want to reach the lost in every community. We want to reach the next generation and make disciples of them and teach them everything that you have commanded. Lord, we want to help marriages be restored. We want to reach foreign exchange students that are on Western Michigan University. We want to raise up missionaries from this house to get on airplanes and go to other parts of the world and preach the gospel. And they also go to the other side of town and they serve their neighbors.  Lord, it's for you. This is what we're called to. Forgive us for the times, Lord, where we've preferred comfort over commission. Lord, forgive me for the times when I've made it about me. Forgive us, Lord, for not being moved by the things that move your heart. We're so grateful that when you saw us in our sins, you didn't leave us the way that you found us. We're so grateful for the cross. We don't want one person to slip into eternity that hasn't had an opportunity to hear the good news of the gospel.  And by God's grace and in unity together, that's what we are going to do.


From the very next day of fire incidents took place, as you all know that our facility burned down. Now we return into a health post with limited services, but we are trying to do our best to provide service to the beneficiaries. Many of know that facilities were burned down, they don't know where to go, so that we are sending our CSW supervisor volunteers, engaging them to deliver the message to the beneficiaries that we are ready to serve, whatever they need. After the fire incident took place to our facility, there is a disruption in vaccination like for the outage program. Right now it is not possible to provide the outage vaccination, but we are with limited services and services, we are providing EPI vaccine, which includes BCG, OPV, IPV, Dipteria, Partosis, Tetanus, Hemopilus, Influenza, Hepatitis B, Miscells and Rubella. We are also providing vaccine for the childbearing mothers, which included two doses, one-month interval. Most of the beneficiaries are struggling with their livelihoods, so the idea of getting vaccination is out of their mind. We are trying to reconnect the bridge to a community health worker to visit door-to-door and encouraging them to come to the facility for vaccination.