So, defining the distribution function where we said that this is equal to F B. So, probability X less than or equal to B. Now, we can immediately write down the formula for the probability of X being in an interval using the distribution function. So, here the idea is that we want to write X between A and B. So, I am taking strict inequality here and so then you see I am writing this set the event X less than or equal to B as a union of two disjoint events X less than or equal to A then union X between A and B. So, X strictly greater than A less than or equal to B. So, these are two disjoint events and they add up to the union is equal to this event then because they are disjoint when I write take the probability it will be the sum of the two probabilities. So, probability X less than or equal to A plus probability A less than X less than or equal to B and so this probability then can be written as difference of probability X less than or equal to B minus probability X less than or equal to A and therefore, this is F B minus F A and you can see immediately that if you want to write equality here then this event would be X strictly less than A. So, in that case we will have to write A minus and so on. So, therefore, since the discrete case it matters whether equality is there or not. So, therefore, this is the way I have written it and then of course, if you want to if you want to also have the probability that X is equal to A then I will simply add to it. So, plus it will be P A right because X equal to A again is disjoint from this set and so I can write here plus probability A and so on. So, you can you know get various forms of the probabilities for X being in an interval and you can write it through your distribution function. So, this was for lecture 5. Now, I will now start defining as since we have already talked about Bernoulli random variable which was a very basic random variable discrete random variable. So, we will define a discrete random variable which is known as the binomial random variable and the notation is b n comma p. So, here n independent trials are performed each resulting in a success or a failure. So, we just say that when you perform a trial then either the outcome is a success or it is a failure. Now, probability of success we say is p and that is why the notation b n comma p that means, number of trials is n and the probability of a success is p and so X the random variable denotes the number of successes. So, in n trials you want to know how many successes have occurred because it is an uncertain event. So, the probability of X equal to r is given as n c r p raise to r 1 minus p raise to n minus r. So, obviously if you are looking for r successes in n trials then they can be any of the r trials out of those n r trials out of the total n trials. So, n c choose r I mean n choose r and then since r successes. So, p into r into 1 minus p raise to n minus r. So, r successes and n minus r failures and this is for r varying from 0, 1 to n. So, for any value of r you can substitute here and get the corresponding probability that X is equal to that r and now you can sum up this all the probabilities. So, that means for r from 0 to n sigma probability X equal to r then this will be summation r varying from 0 to n n c r p raise to r 1 minus p raise to n minus r and this you can see is the binomial expansion of p plus 1 minus p raise to n. That is another reason why it is called the binomial random variable. So, when you write p plus 1 minus p raise to n this adds up to 1. So, therefore, this is a PMF and of course, so the X is the binomial random variable with probability mass function defined by 1. So, now let me again emphasize this fact that when the random variable is discrete we describe its probability mass function. That means when it gives you the function which specifies the probabilities for different values of the random variable X then that is known as probability mass function and when we talk of continuous random variable that means when the random variable can take all possible values in a interval then it the function which describes the probabilities of the random variable that is known as probability density function. So, I will try to keep this thing always in mind, but sometimes may be it may happen that I have random variable is X and instead of saying the probability mass function I may have use the word probability density function, but remember that we make this distinction that for discrete random variables it has to be probability mass function and for continuous random variables it is the probability density function. So, the notation is b n comma p. So, that means a binomial and you only need these two parameters what is the number of trials and what is the probability of success. And so now you want to verify that this actually is a valid pdf and for which for this verification I will need to say that all probabilities must add up to 1 because X can take any of these n plus 1 values. So, the sum of the probabilities for all these values must add up to 1 and you see that this is nothing but the expansion of p plus 1 minus p raise to n the binomial expansion. And therefore, this number is 1. So, 1 raise to n is 1. So, that verification is straight forward. So, then I immediately want to compute the expected value of this random variable. So, that will be because this X equal to r. So, r into n c r p r 1 minus p raise to n minus r are varying from 0 to n. So, now here because this is n c r. So, you will have in the denominator may be I can use the. So, here if I take n outside see the thing is that n c r you can write as n factorial upon r factorial n minus r factorial. So, r if I take out then this will be r minus 1 factorial. So the r cancels out and n I have taken out from here. So, this will become n minus 1 factorial.  And therefore, by taking out n here and cancelling out the r I get n minus 1 choose r minus 1. And since you see here in this summation r is from 0 to n and when r is 0 this term is the first term is 0. So, therefore, no contribution. So, therefore, this summation can as well be written as r from 1 to n and this is what you have.  So, this n p is outside this n is outside and then you have this. So, now this you can see is again a binomial expansion of p plus 1 minus p raise to n minus 1 because the powers are all n minus 1. So, therefore, this is 1 and so your expected value is n p. So, again a straight forward computation you should be able to figure it out. And we can also say that a Bernoulli random variable is binomial 1 comma p. It is a binomial random variable with parameters 1 and p. Let us look at this example taken from Sheldon Ross. So, it says that it is known that screws produced by a certain company will be defective with probability 0.01. So, the p is given here. So, here defective we are treating as a success. So, the probability of getting a defective screw manufactured is 0.01 and these of course, the defectiveness of the screws is independent of each other. The company sells these screws in packages of 10 and offers a money bag guarantee that at most one of the 10 screws is defective. That means if more than one screw is defective in a package of 10, the company will take back the package and refund the money. So, the question is what proportion of packages sold must the company replace? So, that means you are wanting to find out the probability of packages of 10 that have more than one screw defective. So, that means you are wanting to find the if x is the number of you say you say that x is the number of defective screws, then what is x is binomial 10.01. So, because we are treating a defective screw as a success. So, therefore, in a package of 10 the probability is 0.01. So, this is binomial 10 comma 0.01 and you want to find the probability that x is greater than or equal to 2 because when x is greater than or equal to 2 the package will be returned back and the money refunded. So, here I have said that where x is binomial this. So, therefore, now this event I can write as complementary. So, 1 minus probability x equal to 0 and probability x equal to 1. So, these two are acceptable and if x is more than 1 then it is not. So, 1 minus. So, these two you can see that fine. Now, x equal to 0 is the probability. So, 10 c is 0 into p raise to 0 1 minus p raise to 10 and x equal to 1 will be 10 choose 1 p into 1 minus p raise to 9 and now your 1 minus p is 1 minus p is 0.99. And so, you write down this number you can use your calculator to compute it it comes out to be 0.004. So, that means 0.4 percent of the packages will have to be returned and this is you know you can see that this kind of calculation is very helpful to a company which is trying to budget its manufacturing of the screws and anywhere situation you want to know that what is the you know of course, nobody is saying that this exactly 40 percent of the packages will have to be returned, but now the company has an idea as to I mean 4 percent sorry 4 percent of the packages have to be returned. So, there is some idea and a guideline for the company to see what can happen. Now, the next thing that we want to compute about the binomial random variable is its variance. So, first we will compute expectation of x square second order moment and so that will be given by r square n c r p raise to r 1 minus p raise to n minus r summation is from 0 to n. Here again you see for r equal to 0 the first term is 0. So, there is no contribution and here I will do the same thing I will cancel 1 r with this one here and then 1 r I have left with p I will take outside and then n of course, also I should have written which I have done here. So, in that case you will be left with r n minus 1 and the same thing same trick I will do and this is it and n minus r I will write as n minus 1 minus r minus 1. So, this is p raise to r minus 1 and this is the whole thing and the r that 1 r that is left here I will write it as r minus 1 plus 1. Now, you see you can break up this sum into 2. So, r minus 1 times this whole thing you see now is the expectation of a. So, if y I define y as a binomial n minus 1 comma p. So, y is a binomial random variable where n minus 1 trials have taken place probability of success is p. So, then you see in this summation I should have written here 1 to n. So, then r minus 1 into this term this whole thing summation r from 1 to n will give you the expectation of binomial n minus 1 comma p. So, therefore, that will be n minus 1 into p. So, n minus 1 into p and plus 1 and then this summation this again is the same thing p plus 1 minus p raise to n minus 1. So, that is equal to 1. So, therefore, you get n p times E y plus 1 expectation of y plus 1 from here and expectation of y is n minus 1 p and this is plus 1. So, n p into this. So, now for the variance you have to write n p into n minus 1 p plus 1 minus n p whole square and when you simplify you get this and some people also write this as n p q where q is 1 minus p. So, easy way to remember and we will see some more applications of and of course, as the course progresses you will continue to see where all you can use the concept of a binomial random variable. Let us look at this proposition about the random variable binomial n comma p. So, then as x goes from 0 to n this number probability x equal to k increases monotonically and then decreases monotonically reaching its largest value when k is when k is n plus see integer part largest integer part of n plus 1 into p see p is a fraction. So, n plus 1 p may not be may not be an integer. So, when we write brackets like this it means that the largest integer for example, if you consider 5 by 2. So, the largest integer here would be 2 largest integer which is less than n plus 1 into p. So, this is this similarly if you consider what shall I say 9 by 4 here again the largest integer will be 2 because 4 times 2 is 8 and then it is 1 by 4. So, this is what we mean by that. So, that means the binomial probabilities they keep on increasing reach the largest value highest maximum and then starts decreasing this is the idea and. So, we want to prove this result. So, let us look write down probability x equal to k upon probability x equal to k minus 1 and. So, if you write down the expressions this is what you get.  So, therefore, you will be left with n minus k this will come in the numerator.  Now, this should be we want to know for what values of k the probabilities are increasing.  So, this gives you this. So, the as long as k is less than n plus 1 into p the probabilities because you see this is probability x equal to k divided by probability x equal to k minus 1. So, this number is greater than this number as long as k is less than n plus 1 into p. So, that means and since k is an integer we will say that this goes on increasing till k reaches the largest integer present in n plus 1 into p. So, that is what we say. So, that means for k less than or equal to this integer part in this number the probabilities are increasing and when k is greater than this because the inequality will get reversed. So, then they are decreasing and if n plus 1 into p is an integer then of course, that will be the maximum value otherwise because here it is strictly less and this is less than or equal to. So, k will the probability x equal to k will attain its maximum value for n 1 into p that means when this is an integer otherwise it will attain its maximum value for the largest integer present in n plus 1 into p. So, I will show you the diagram now for a particular this thing the bar chart for a binomial distribution. So, here we will look at the bar chart when the random variable is you know the parameters are 10 comma half that means your p is half and the number of trials is 10. So, you see here if you look at this thing 1 by 2 that means for x equal to 0 then when k is equal to 0 the probability p 0 is 1 by 2 raise to 10 and that is depicted by the small bar here and then x equal to 1 it starts to increase and then at 5 it is maximum. So, if you look at see we said that. So, what is your n plus 1 into p this is equal to 11 into 1 by 2. So, this is 5 1 into p 1 by 2. So, the integer part is 5. So, as just now we looked at the we proved this result that in this case the maximum value will be attained for the largest integer present in n 1 plus n plus 1 into p which is 5. So, therefore, you see the largest bar is corresponding to k equal to 5 and then again the value start decreasing and you also see that in this case because p is half. Therefore, the graph is symmetric about the that means about the value x equal to k equal to 5. So, we just want to explain that this is 1 0 2 4 into p raise to k because you know the horizontal axis is k and here just to otherwise the numbers would have been because the probabilities are less than 1. So, therefore, just to make them whole numbers we multiplied everything by 1 0 2 4 which is 10 raise to 2 raise to 10. 2 raise to 10 if you multiply all the numbers all the probabilities then they become integers. So, this is what your binomial bar chart will look like for the particular value when p is equal to half and I have shown you also that how it will continue to the probabilities will continue to increase reach a maximum for the integer part of n plus 1 into p and then start decreasing. Now, let us look at the distribution function of the binomial random variable that means you want to compute probability x less than or equal to i and here you see the expression would be you have to sum up all these probabilities from 0 to i for the individual probabilities of the random variable takes. Now, these numbers can be very very large. In fact, if your n is moderately large and you know you want to compute even for n equal to 20 and if you want to compute for let us say i equal to 16 then you know you will have to add up 17 terms here and this can become very tedious. So, there is a very simple and nice formula which you can recursive formula because I we obtained this here in this expression I have this here right.  So, in that case from that formula you get this recursion. So, which says that if you have obtained probability x equal to k then you can obtain probability x equal to k plus 1 by the simple formula recursion formula right and you can see that to start off when x is equal to 0 when x takes a value 0 then this will be 1 minus p raise to n. So, once you have computed this then probability x equal to 1 will be from this formula p upon 1 minus p k is equal to 0. So, this is n and p x equal to 0. So, therefore, this will be probability x equal to 1 will be p upon 1 minus p n 1 minus p raise to n. So, that means if you have already computed this number then you have to simply multiply the probability x equal to 0 by this number to get probability x equal to 1 and then similarly probability x equal to 2 would be this number multiplied by probability x equal to 1 which you have already computed here. So, a nice simple recursive formula that you have obtained and you can write as computer program feed the values and then it will go on computing that means you have to just feed the value n and p and then it will compute the successive probabilities for you. So, this is you know the computational part because otherwise things can become very tedious. Now, another special kind of discrete random variable is the Poisson random variable and this is named after the mathematician S. D. Poisson who defined this random variable in 1837 and he in fact wrote a book which was application of probability theory to lawsuits, criminal trials etcetera. So, you see the kind of applications that the Poisson random variable has and at that time in 1837 he wrote this book where he applied the theory of Poisson random variable to you know predicting things about lawsuits and criminal trials etcetera. So, x is a random variable which takes value 0, 1, 2 up to infinity. Lambda is the parameter and lambda has to be positive. Then we define the probability that x is equal to r by this number e raise to minus lambda lambda raise to r upon r factorial defines a probability mass function of the random variable x and this is a Poisson PDF and now you want to make sure that this defines a PDF. That means you have to show that summation probability x equal to r sorry x equal to r is 1. So, we will say very simple calculation will immediately give you the verification that this is indeed a PDF. So, to show that the probability mass function defined for a Poisson random variable is valid PMF. We add up all the probabilities here which is equal to this, but you see lambda raise to i upon i factorial i varying from 0 to infinity is nothing but the expansion of e raise to lambda. So, therefore, e minus lambda into e raise to lambda gives you 1 gives you e raise to 0 which is 1. So, we have checked the validity and of course, each of the probabilities defined are also non-negative. So, therefore, what we have defined as the probabilities for the random Poisson random variable forms a valid PMF probability mass function. Examples of random variable that obey Poisson probability law some of the situations I am just writing down to get give you a better feeling for the this random variable and this is for example, number of misprints on a page or a group of pages of a book which we believe is a random phenomena. Number of people in a community living up to the age of 90 which again is a random phenomena right because how long one is for how long one lives is not a certain event. Then number of wrong telephone calls dialed in a day or wrong numbers what I mean is wrong telephone calls dialed in a day. This could be at a particular exchange or you may take a particular number and then count the number of times the wrong telephone calls come. Number of customers entering a post office in a day then number of alpha particles discharged in a fixed period of time from some radioactive material. Now, you see that here in all these situations there is some sort of discreteness and that is why we are saying that these situations can be modeled by a Poisson random variable. You can just get the feeling because you cannot have two numbers dialed simultaneously there has to be a gap right. Then number of misprints of course will be you know occurring discreetly some gap of time and so on. Now, I want to show you the relationship and as we go on the discrete random variables that I define I want to show you the relationships between amongst these discrete random variables. So, Poisson approximation of the binomial random variable and this is when n is large. So, for n large and p small then you expect that n p would be a very moderately small number n p and so we define that as lambda. So, in other words what you are saying is that p small n is large and then this n p number sort of approaches reasonably small number equal to lambda right. Then you now we will let us look at the binomial probability for x equal to i and x is equal to i then this is the probability defined then let me start writing. So, here if I take this then p is lambda by n. So, I will make that substitution here lambda by n raise to i 1 minus lambda by n raise to n minus i then if you cancel out the n minus i factorial part here you will be left with n into n minus 1 n minus i plus 1 and then this n raise to i I am writing here and i factorial is underneath because you see I am trying to converge to the Poisson probability. So, lambda raise to i upon i factorial and this one 1 minus lambda n raise to n and then divided by 1 minus lambda n raise to i. Now, this n 1 n cancels and I am left with n raise to i minus 1 and you will have i minus 1 terms here. So, I take n inside and divide. So, therefore, each of this term becomes 1 minus 1 by n 1 minus i minus 1 by n right. So, this n raise to i gets absorbed here lambda i raise to i factorial 1 minus lambda n raise to n and this. Now, as n goes to infinity becomes very large then 1 minus lambda by n raise to n will approach e raise to minus lambda. So, I hope you are all familiar with this limit and then 1 minus lambda by n raise to i because as n becomes large lambda is fixed. So, this number is a becomes smaller and smaller and so this will approach 1 for n sufficiently large right. And then right and all these numbers again as n goes to infinity or becomes very large each of these numbers approach 1. So, their product is 1. So, therefore, this is gone this is gone and you are and this raised goes to e raise to minus lambda. So, the whole thing approximates to lambda raise to i upon i factorial e raise to minus lambda which is the Poisson probability. So, essentially I should say that this approaches this and this is your. So, these things are actually interlinked and therefore, for large n your binomial probability is the same as the Poisson probability that is one result. Now, computing the expected part expectation of a Poisson random variable again straight forward sigma 0 to infinity i lambda i e raise to minus lambda upon i factorial. So, here I do the same trick as I did for the binomial. So, lambda e raise to minus lambda you take outside then your i becomes the summation becomes from 1 to infinity lambda i minus 1 upon i minus 1 factorial which is again the expansion of e raise to lambda. So, lambda e raise to minus lambda into e raise to lambda it gives you lambda. So, the whatever the parameter of the Poisson distribution that is also its mean. So, this is one result and now variance also straight forward again the same thing I will do as I did for the binomial expectation of x square we find out then again the i cancels lambda you can take outside and the i that you left you write it as i minus 1 plus i and this thing and just do the same argument as we did for the binomial you will see that i minus 1 this will give you the again a Poisson this thing expectation of a Poisson with lambda parameter. So, this will be lambda and plus 1 because sorry this should be plus 1 i minus 1 plus and 1 because these are again Poisson probability. So, we will add up to 1. So, this is what you have and therefore, variance will be given as lambda into lambda plus 1 minus lambda square which is the expectation. So, this is a particular situation where your expectation and the variance are the same and both are equal to the parameter of the Poisson distribution. Now, best way to give you feeling about particular random variable is always through examples. So, here let us look at the Poisson error model and as I said this is same the first one there that is you are modeling the number of printing mistakes done by let us say high speed printer. So, here it says that certain high speed printer makes errors at random on the printer page on the making an average of 2 mistakes per page. So, on the average 2 mistakes are made per page assuming that the Poisson distribution with lambda equal to 2 is approximate to the to model the number of is appropriate actually I should say I should say appropriate let me correct the word appropriate. Appropriate to model the number of errors per page what is the probability that of 10 pages produced by the printer at least 7 will have no error. So, now you want to find out the probability that when the 10 pages are printed 7 of them are without any error. So, we assume independent of error from page to page that means the number of errors that occur in one page is independent of the number of errors that occur on the second page and so on. Now, let X be the number of errors on a page then as we have said that we will model this by the Poisson random wearable and so probability X equal to R will be e raise to minus 2 into 2 raise to R upon R factorial where R can vary from 0, 1 to anything and we said that since the average number of errors made by the printer on a page is 2. So, I am taking 2 as the parameter of the Poisson random wearable that I am using to model this particular situation. So, this is what is given to you now if you want the probability that a page is error free that means there are no errors. So, probability of X equal to 0 is equal to e raise to minus 2. Now, the thing is that you want to find out the probability that at least 7 pages are error free. So, you see we modeled the situation by Poisson random wearable to find out the probability of a page being error free. So, that I got as 0.1353 which is equal to e raise to minus 2, but now there is a next step and see that is why the example is very interesting because you see now you will make use of the binomial random wearable because having no error on a page is a success. Let us treat that as a success and since we have said that and we are looking at 10 pages. So, I will consider scanning every page as a trial of the experiment and if there is no error on a page then that will be a success. So, this is ideal for a binomial random wearable this situation is ideal because since pages are the errors from page to page are independent. Therefore, I will treat this as 10 independent trials and if a page turns out to be error free then that is a success and what is the probability of a page being error free that is 0.1353. That means the probability of a success our P is 0.1353 and the number of trials is 10. So, this is a binomial random wearable and so probability that at least 7 pages are error free I require that 7, 8, 9, 10 at least 7. So, the error free pages can be 7, 8, 9 or 10 and the probabilities will be 10 choose 7 e raise to minus 2 raise to 7 into 1 minus e raise to minus 2 raise to 3. Similarly, for 10 choose 8 we will write the expression e raise to minus 2 raise to 8 and so on. Then 10 choose 9 e raise to minus 2 raise to n into 1 minus e raise to minus 2 plus 10 choose 10 which is e minus 2, 10 and even as I was saying that here also the computations have to be done by using a calculator or a computer. So, I write down the numbers this is into 10 raise to minus 4 and so the probability is 0.0007. So, which is very low. So, therefore, what you will say is that the probability that 7 pages out of 10 will be error free is a very, very low probability has very low probability. So, the chance of this happening is very small. Now, again I will try to take as many examples as possible. So, make the concepts clear. So, first this example this question I am taking because that will lead to a Poisson approximation which I want to show you. So, first consider the situation here that n people are present in a room. What is the probability that no two of them celebrate their birthday on the same day of the year? So, this is the first question what is the probability that no two people will be having a same birthday celebrated on the same day. And then we want to ask the question how large need n be so that the probability is less than half. That means how many people should be there in a room so that the probability of any two of them having the same birthday is less than half. We want to first look at this question and then I will take you to the how I approximate this situation and give you the you know through Poisson and then again show you the connection between the two. So, the birthday problem as we saw that if no two of them have the same birthday then of course we will write it as 360. See the first person can have a birthday on any of the 365 days and we are of course ruling out the leap year. So, nobody has a birthday on 29 February. So, this is 365 upon 365 then the second person can have in the group of n people. So, second person can have his or her birthday on any of the remaining 364 days and so 364 upon 365 and so on. So, this will go on up to 365 minus n plus 1 upon 365. So, this gives you the probability that no two of them have the same birthday and we want the number n such that this probability is less than half. So, the probability that no pair no two people in the group have birthday on the same day we want that probability to be less than half and we want what should be the smallest number of such people. So, that this probability is less than half and so it turns out that n is n greater than or equal to 23 satisfies this inequality. Now, and of course then I started calculating backwards. So, for example, for n equal to 23 for n equal to 23 365 minus n plus 1 is 343. So, then I started computing 343 upon 365, 344 upon 365 and so on and then just 363 upon 365 when you come up to this point then it probably just turns less than half. So, the next number is 364 by 365 which is also less than 1. So, the probability will remain less than half and of course the last number is 365 upon 365 which is 1. So, therefore and in fact if I took n to be less than 23 that means if I took n to be 22 then this will not happen because then you will be starting with 344 and then when you calculate backwards it will not turn less than half. So, this is just enough and that means if you have more than 23 people in the group then the probability will be still less than half. So, this is the idea. Now, here calculating it otherwise becomes difficult because what is happening is that you are in the group if a and b have the same birthday then it is possible that b has the same birthday with c and so this thing is transitive and so it is not very easy. I mean one can not really assume independence to calculate the actual probability, but we will see that some approximation is possible and so we will see that, but so this is just interesting to see that even in a group of people having 23 I mean a group having 23 persons the probability that no two people have the same birthday is less than half. So, let me continue with the Poisson approximation of what we discussed with the problem the common birthday problem we discussed. So, now suppose choosing a pair is a trial choosing a pair of people present in the room then there are n c 2 pairs that I can different n c 2 pairs that I can choose. Now, consider e i j as a trial i j i not equal to j such that and of course, what I mean by this is that I am choosing the i th and the j th person here i and j are different having the same birthday. So, e i j would be considered the event when when I choose the pair i j it is a success that means they have the same birthday. Now, in question 7 of the exercise 3 which I will be discussing at the end of having gone through all different kinds of discrete random variables then in. So, this exercise 3 question 7 I am asking you to show me that these pairs the events e i j are pair wise independent. So, of course, choosing of the choosing of each pair is independent and then I want to show you I want you to show that the pairs e i j that means the set of events e i j i not equal to j that means n into n minus 1 by 2 such events they are all pair wise independent which means that i j and e l k yeah this is important I must show i j and e l k where i is not equal to l and j is not equal to k. So, that means different sets of pairs when you choose different sets of events I am sorry that is what I should say that when you pick up 2 such events from here where i is not equal to l and j is not equal to k then any 2 such pairs are independent events this you should do as an exercise at the end of this chapter. Now, x is the number of successes in n c 2 trials having the same birthday. So, let me now start looking at the Poisson approximation. So, number of successes in n c 2 trials having the same birthday and this I am calling as and so as I discussed with you is a binomial n into n minus 1 by 2 comma 1 3 1 upon 365 remember I showed you that pair of people having the same birthday the probability is 1 upon 365 and the number of trials that I have is n into n minus 1 by 2. So, this is a binomial random variable if I do the Poisson approximation then we said that the corresponding Poisson parameter would be n p and so that will be n into n minus 1 by 2 into 1 upon 365. So, this is my n p and now you want to compute the probability that no 2 people have the same birthday that means you want to find out probability x equal to 0 which is e raise to minus n into n minus 1 by 2 into 1 upon 365. So, which is this number. So, the next part of the question was what value of n satisfies this the inequality that this is less than half no 2 people having the same birthday that probability should be less than half. So, if you take the logarithm of both the sides then this is minus n into n minus 1 upon 730 l n of e is 1. So, this is less than minus l n 2 so minus minus cancels the inequality reverses. So, therefore, n into n minus 1 is greater than 2 into 730 and you can see that n equal to 23 will satisfy this equation. And therefore, this also is validated the answer that I got earlier by other arguments now I have been able to validate it by. So, therefore, this is what through these examples I am trying to show you that you know how you model and sometimes your modeling can be such that the getting the answer can be very cumbersome. But sometimes when you model it in the right way then you can get the answer. So, here the computation was quicker then because remember there I was solving the I was trying to say 365 into 364 up to 365 minus n plus 1 divided by 365 raise to n this should be less than half. And so I was wanting to compute an n and you could see that you know this will require trying out trial and error of lot of values of n before you get the answer and we got the answer I showed you that the answer would be 23. But now if you model the situation through Poisson random approximation of a binomial through Poisson then you get the answer in a much simpler way.


You don't need to be inspired or motivated to go take a sh**. Today, we're going to talk about the action steps that you need to take to quickly get out of a rut. And I'm going to give you seven ways, seven action steps that you can use to get out of a rut. And to start off today, I want to give you a phrase that I say all the time to kind of help you understand what a rut is and how to get out of it. And that phrase is this. Inaction causes more inaction. Action causes more action. What do I mean by that? If you're in a rut, you're never in a rut when you're moving. Like if you ever feel like you're stuck in a rut, it's never when you're moving. It's always when you're stagnant. And I always use the phrase stagnant on purpose because I think very visually, when I think the word stagnant, I think of like a pond that's been stagnant for a long time. It's got like a gross film over top of it. There's flies. And it's just becoming disgusting because I never want to think of my life as stagnant as that pond is. And I get it. If I'm saying you have to take action in order to get out of a rut, you're like, well, what the hell? Like I'm trying to learn how to take action, right? That's the reason why I'm in the rut is because I can't force myself to take action. I get it. So I'm going to give you my seven tips to actually get out of a rut. And I get it. Like you take action all day long though. So you might be sitting there like, yeah, but Rob, like I, I'm not, I want to take action, but I'm not taking action that I want to, right? So I'm just, I'm not taking action. So how am I supposed to take action when I'm not taking action? You have to realize you're taking action all day long, right? If you're in a rut, you're not just sitting on the couch and then peeing on yourself, right? Getting up off of the couch, walking to the bathroom and peeing is action.  Both of those things are taking action. The only difference is it's not the action that you want to take for the life that you want, right? You don't need to be inspired or motivated to go take a s***. You're going to do it if you have to, right? You don't have to do it to go pee. You don't have to do it if you're hungry. And I'm telling you this because I want you to actually really see this and feel this in your mind. You're tricking yourself into thinking that you're stuck in a rut, right? If a lion walked into your house right now, you wouldn't have to search for motivation to run from it. You wouldn't, right? You would just get moving. You'd run. You'd take off, right? And so you have to realize you're tricking yourself into thinking that you're in a rut in the first place. And I'm going to give you obviously steps to get out of it. But what it's really about more than anything else is starting small more than anything else. A lot of times you're not taking action because you're thinking about all of the things that need to be done versus what's the next thing that needs to be done. You're thinking about all of the things, right? If you want to go to the gym and you're not motivated, you're stuck in a rut physically and your body's not where you want it to be and you know you need to go to the gym. It's not because of the fact that you have to go to the gym and do a curl. It's because of the fact that you're thinking not of that one curl, you're thinking of every single exercise, every single thing that you need to do. Getting into the car, well, first off, getting dressed, getting yourself ready, making sure that you have water, putting on your shoes, getting into the car, going to the gym, lifting the weights, doing every single set that you and you start thinking about all of them versus what's the next thing they need to do, right? You're not thinking about one curl, you're thinking about all of the curls, all of the sets that you need to do, all of the reps that you need to do. And sometimes you have to take it back to just the simplest form, right? There's two things you need to focus on. Number one is your mind and number two is your body, right? In order to get out of a rut, you need to trick or get two things moving. You need to get your mind moving and you need to get your body moving, right?  I can do this. Maybe it's David Goggins, you need someone to yell at you and scream at you and cuss at you or maybe you need someone inspirational or motivational or someone spiritual. I don't know what it is, but you need to trick your mind in a way to get your mind actually moving. So that's the first thing you gotta think about. What can you do to get your mind moving? Second thing you gotta think about is your body. You just have to physically move your body. Once again, I said, if you have to go to the bathroom, you're going to get up and go to the bathroom stuck in a rut.  If you're hungry, you're not going to starve yourself. You're going to get up and you're going to take small action.  So motivation can come from the mind, motivation can come from the body. So that comes to the very first thing. So out of the seven tips I'm going to give you, number one is to move your body. Whatever it is you need to do, do not negotiate with your mind. If you feel like you're stuck in a rut, nobody's stuck in a rut moving. Like I said, you're always stuck in a rut and you don't feel right and you don't feel motivated when you're sitting on the couch and you're looking at Instagram. Hey, delete Instagram already. Get rid of it if you have to. If it's getting in the way of holding you and holding you back from the dreams that you want, get rid of it, right? Move your body. If you're not moving your body, you find it hard to move your body. I need you to get enough motivation to get up and get some form of caffeine in you if you need to, something that's going to get you moving. If you want something that's that's not caffeine, you can do B12, right? Take vitamins, whatever it is that you have to do to get your body moving. What happens is once again, inaction creates more inaction. Action creates more action. So if you go, you know what? I don't feel 100%. I don't feel ready to do this. All you have to do, force your body to move, get up, do some jumping jacks, put on some music, whatever it is that you have to do, right? Take small steps, which is number two. So number one is get your body moving. Number two is to take small steps. Once again, as I said earlier, if you're not taking action towards the life that you want, towards the things that you need to do, it's because you're thinking about doing all of the things that you need to do even though there's no possible way that you can do all of them at one time, right? Just think about, okay, I need to get in shape. All I need to do is go to the gym and just do one curl. That's it. I don't need to do anything else. I just need to do one curl. And when you get to the gym and you do one curl and just make that the only thing you need to do, don't even, you don't have to do anything else. If you go into the gym and you do one curl and walk out, that's a win because you've at least gone in the right direction of what it is that you want. But I guarantee you when you get to the gym and you do one curl, you have a weight in your hand, you're like, you know what, I might as well just do a couple more and you do a couple more. And then what happens is your body starts to create endorphins. Your body starts to create all of the feel good chemicals that I'll talk about, right? What you do is your body will trick itself into going, you know what, this actually feels good. Maybe I should keep doing this. And then you do it again and you do it again. And then you realize 45 minutes later, you're not focusing on the whole 45 minutes. Realize that you're focusing on just the next curl, just the next squat, just the next foot in front of the other in your walk on the treadmill, whatever it is you need to do, take the small steps. Don't worry about everything. Just worry about what the next thing is in front of you. So that's number two to get out of a rut is to take the small steps number three. This is something that not enough people talk about. Celebrate your small wins. If you got up, you put on some clothes and you got in the car and you got to the gym and you walked inside of the gym, that's a win, right? You've got to give yourself celebrations for all of these little teeny tiny micro wins, right? Dance after you get that first set done. You know, if you're, I have a gym in my garage, right? If I don't feel like it, I'm going to put on the music. I'm going to drink some caffeine and I'm going to make myself feel really good. I'm going to dance around in my garage and get my body moving. And I'm going to celebrate the small wins. This is the reason why this is important. Number one, it gets the body moving. You get on the good music and the good music always makes you feel better, but your brain will release serotonin and dopamine and serotonin and dopamine make you feel good. And when you feel good, your body and brain want more of those feel good chemicals. How do you get the feel good chemicals by celebration? That's when they're released.  I'm like, I just got to force myself to get up and do it. I forced myself, I get my clothes on. I walk into my garage and I'm like, I'm just going to lift one thing, right? I lift one thing. That's all that it is. I'm like, all right, I did what I needed to do. Let me put on some music. Let me celebrate this success. And when I celebrate and I move around and I dance and I start to force myself to feel good, which is something that is 100% possible for every single person right now is to force yourself to feel good by having little celebrations. I go, whew, I actually feel really good. You know what? I actually could do another set and I do another set and I move around and make myself feel good. I put on some of my favorite songs. I put on my feel good music. I put on my happy playlist and I do the next set and I do the next set. I celebrate every single one of them. And what happens is it's starting, I start to get momentum on my side, right? Which is, I don't think I'm going to talk about in just a minute, but I'm not there yet. Number four is to turn off your phone. Yes.   Getting on Facebook, getting on Instagram, like all of those things are great. I think that they're great if they're used correctly, but I think they're misused by most people. So turn off your phone. Then what do you need to do? Turn off all of your electronics, right?  I'm just going to sit here. You're going to get bored and you're going to go, I have to have something to do. Why? Because humans at this point in time, we're so addicted to doing things. It's really hard to just sit on your couch and stare at the ceiling. Trust me, I've tried it, right? I do it sometimes. I force myself to literally stare at the, at the ceiling. I'm like, I'm not going to look at my phone. I'm not going to look at electronics. I'm not going to do something right now. I'm going to literally just sit and be, but for most people, you get to that point. You're like, I got to find something to do. I've got to do something and you get bored because your phone is off, your TV is off, none of this. You can't turn your computer on. You can't go on to YouTube, TikTok, Instagram, Facebook, none of those things. You'll get bored and you'll go, I got to do something. Guess what? That's what you want. You want to be able to do something. So literally you've forced yourself into a mini boredom, which is then going to make you then take action the way that you want to. So then what do you do? Okay. I'm in this mini boredom. I feel like I want to do something. Number five is focus on just the next 60 seconds. I don't care about the next five minutes. I don't care about tomorrow. I don't care about the next hour. I don't care about later on today. I don't care about this afternoon, this evening, any of those things. The next 60 seconds, if I'm on the couch and I'm bored because my, my, I've, I got no Instagram. I got no Facebook. I got no TikTok. I got none of those things that come and save me. I can't turn on Netflix because my TV's off.  I take all of those things away. I go, okay, I want to do something like I feel like I should do something. What do you do? Okay. In the next 60 seconds, what can I do to get me closer to this goal that I want that's going to get me out of a rut of, you know, the body that I want? For instance, if we keep going down that route, what can I do in the next 60 seconds? I can do 20 push-ups. All right, let's just do 20 push-ups. And that alone, the next 60 seconds, that's going to take me into the direction that I want to move with my life. That's the key. What can I do in the next 60 seconds? It's going to move me into the direction of where I want to go with my life. And you can ask yourself, this question is what I'm doing right now, getting me closer to or further for my goals. Is what I'm doing right now, getting me closer to or further for my goals. Okay. So then of those next 60 seconds, boom, 20 push-ups. Maybe I put on some feel good music. Maybe I'd celebrate a little bit. Maybe I get all excited. I get that happy feeling. I get those dopamine, the serotonin, all of those things. I start to feel a little bit better. All right, cool. I'm starting, I don't feel like amazing, but I'm starting to feel a little bit better. You know what I'm telling you, you've done this before, right? You start to feel, all right, maybe I could do another set of 20 push-ups, whatever it is. All right, that brings us to number six, which is ride the momentum. Now I'm starting to feel just a little bit, maybe not a whole lot better, but just a little bit better. I'm going to celebrate. I'm going to get that serotonin. I'm going to get that dopamine. I'm going to get the music playing. I'm going to get the dancing going. I'm going to get the celebrating going. Now that it's rolling, I want to keep it rolling. And this is the important part, is to make sure that when you start to feel the momentum, you don't let the momentum stop. And you ride the momentum until you're done, right? I always say that if you're at the very top of a really big hill, and you take a 15-pound bowling ball, and I decide that I'm going to let the bowling ball go, and I let it go, and then I immediately stop it, it's very easy to stop if I immediately stop it or a half a second later, I stop it. Why? Because the bowling ball hasn't got any momentum on its side. It hasn't got any momentum going to where it's trying to go, to where it needs to go. But if I'm at the top of a 400 foot hill and I'm like, all right, now I'm going to try to stop it at the bottom of the hill, that 15-pound bowling ball weighs a whole hell of a lot more. It feels like it weighs a whole hell of a lot more because it's got so much more momentum. You're not going to reach out your hands and try to stop a bowling ball at the bottom of a hill. It's going to break some fingers. It's going to break your hand. Something's going to happen. And so when you start to feel the momentum, ride the momentum because it's very easy to stop the momentum at the top of the hill. You want to at least start to get down a little bit, a little bit down the hill, a little bit down the hill so it starts moving a little bit quicker. So once I start getting the next 60 seconds and I go, okay, what's the next 60 seconds? I can, I start to feel a little bit better. What's the next 60 seconds feel like? And I'm pushing myself just a little bit more. Right? And then if I want to, to add on top of that, one of the best ways to do that is tip number seven is to get around other people, right? Now, when I say get around other people, I mean, get around the right people, right? You got to make sure you're getting around people who are actually going to be holding yourself to a higher standard. Earlier in the week on Wednesday, I work out with a group of men. There's like 15, 20 of us that work out every single Wednesday. It's just the way that we've been doing it now for the past three months, right? This Wednesday, my alarm went off. I did not want to wake up this Wednesday. Didn't feel like it. It was really freaking cold outside. We work out outside and I was like, no way. I don't want to do it.   This is a little voice that's trying to keep me from all of my dreams.  And when I hear the little inner bitch and I can identify it, I've got to do the exact opposite of what it says. And I was like, all right, I got to get up. I go and I didn't want to work out. But I knew that if I was around 15 to 20 other guys that were working out, it would start to make me feel better. It would start to give me motivation to be around those people that are also doing something amazing with their life. So I forced myself. I was like, I'm not going to negotiate with you. I got some coffee, got out the door, immediately went there. And we worked out for an hour and 21 minutes, according to my heart rate monitor, and I burned almost 1,100 calories over 1,000 calories in a workout. I would not. I promise you this. I would not have burned 1,000 calories. I don't know if I've ever burned 1,000 calories in a workout by myself, but I burned 1,000 over 1,000 calories in that hour and 21 minutes. I definitely wouldn't have gone for an hour and 21 minutes either. But the thing is when you're around other people with the same mission that have momentum going on their side, there's something that happens cohesively when everybody else is working towards the same mission. So are there other people that are working towards getting better in their mind, that are working towards getting better in their business, that are working towards getting better in their relationship, they're working towards getting better in their finances, that are working towards getting better in their bodies, whatever it is that you can get yourself and even if you don't feel like it, you can place yourself around them and being around them motivates you to be better. That's what you've got to think about. Hey, thanks so much for watching this video. If you want to learn even more about Master Your Mind, click right here and watch this video as well.  Failing at something is better than doing nothing.


Hello, everyone. Today, we are going to talk about the theory of Karma. Now, we have started talking about Ethics as practiced in the Indian tradition, and today we talk about a very fundamental notion of this theory of Karma in the tradition of Indian Ethics. Now, many of us must have already heard of this theory of Karma or theory of Karma as said in an as variously said. Now, we perhaps may have an elementary idea of it. Now, let us look at it, what could be the motivations and what is this basically this theory of Karma? Well, as most of us would put in that well, theory of Karma is about, simply put, is that as you so, so shall you reap, that there is a moral equilibrium, that every action has a desert. Now, if you look at the screen, we would talk about this, that well, righteous actions are rewarded, wicked actions are punished, that there is no exception. Well, that is essentially the theory of Karma as commonly understood. Now, let us look at these, look at the motivations for this action. Now, many of us may believe that, every act sticks on to us and we need to, we get a result of these acts and many of us would believe that well, there is a predominant chance factor in the entire gamut of moral actions. Now, let us say, what could be the motivations of the evolution of this theory of Karma? Now, theory of Karma is followed in Jainism, Buddhism and in various schools of Indian philosophy. Now, let us say, let us say of all the children born on the day you are watching this video, that is today, will all of them have an equal opportunity or do all of them have equal access or will all of them be having equal access to facilities in their future or all of them born equal? Well, perhaps without too much of effort, we can answer that well, no, definitely not. Some would be advantage, some would be disadvantage, some will have a lot of advantage, some will have a lot of disadvantage and there will be a variation in the opportunities that each of the individual born on this day face. How do we rationalize this difference in the start line? Now, perhaps our intuitive idea of fairness is that well, the start line has to be the same and whoever performs better, deserves better. But, what if the start line is not the same? Now, entire gamut, our entire gamut of public policies, governmental intervention, philanthropy, social activities are towards getting this discrepancy in the start line adjusted. Now, theory of karma in this sense brings about, is a derivative of what in Indian philosophy is known as Ritta. Ritta or the theory of cosmic order that everything is regular, ordered and there is no chance factor or nothing spontaneous. Ritta is a fact, another fundamental claim of the Indian thinking tradition that there is a cosmic order that nothing is random, nothing is by chance. The key word that we would like to perhaps take here is random, nothing is random. So, well when we say that nothing is random, Ritta or cosmic order says that well whatever happens is a part of a grand design that the entire cosmic order is an order, not a accident, not a chaos, not a play of chance. Now, the theory of karma is a subset of this claim of Ritta or cosmic order that well whatever happens, happens at its cost and there is no random or uncaused events in the moral equilibrium. Now, coming to well let us say that well what does theory of karma do, theory of karma now is claiming that well this discrepancy that we have talked about in the starting line is well not a discrepancy at all. In fact, the discrepancy in the starting line is if analogically compared to that of a group of runners running a race. So, each lap the advantage that you get on you transfer it to the next runner and that is the starting line is not when you are born, but the starting line is way before that that where you are born is a consequence of your past actions. Now, what does this do, this might perhaps seem illogical or irrational to it, but then this is just a theoretical model and how does this model seem to be better because it attempts to justify differences that there is there are discrepancies and these discrepancies are not random, but then perhaps many of us would be see thing with the question that well what does this mean that we that these differences are that these discrepancies are that we do nothing about it. Well we will talk about this in detail, but let me give you an answer from what the karmic theory would say is that well the environment or the situation that we face is a result of our past actions, but what we do with it and what others do with it is where our free will is. So, human actions are not completely determined in that way if the theory of karma is giving an idea that things are determined no well things are not determined, but the trajectory incidents are perhaps influenced by what is your karmic desert and thereof you decide and thereof how you tackle them accumulates fresh desert or not, but that will be towards a later questions of karma. Now let us classify human actions now let us say human actions when taken from the perspective of desire. So, as classified there are Sakam and Nishkama karmas which would mean that well actions which are done out of desire or with desire and desire less. Now again classification of human actions from the point of the effect that it yields can again be made human actions the first classification that is made is Prarabdha and the second classification is Anarabdha. Anarabdha is further classified into Sanjita and Sanjyaman. So, what this basically means is well Prarabdha means that the results are already effectuating Anarabdha is not yet effectuating Sanjita is actions are being done effect due and Sanjyaman is well actions are being done. Let us say what is the point of these classifications now if you look at human actions from the point of desire when we talk about desire we talk about actions which are done with a result or with a goal perceived when we talk about Nishkama karma or we talk about desire less action is when an action is performed without the goal of the action in mind. Now if you look at the other classification of human actions they are that is effects. Now Prarabdha is when which we have we might have heard in common talk before strictly means that we these are actions which are which have been done and their effects are already in process that is they are effectuating. Anarabdha in contrast to Prarabdha is when the effects have not yet been have not yet begin to unfold that is it is not yet effectuating that can again be divided into Sanjita that is the actions that have been done but the effects are due and Sanjyaman where in the actions are being done. So these are basically classifications again to detail the theory of karma that how do we find that well actions yield itself. Now if we take a look at how this scheme of things is fitting into the way we lead our lives is that well every act that we do cannot be without a trace every act or every moral act that is done sticks to the agent. Now this is quite profound understanding because not only a that it perhaps makes one almost fearful of the result of immoral actions and expectant of the good results of moral actions. So theory of karma would in that sense bring about a moral order more out of the fear of punishment and the greed for reward. But is that all that is to theory of karma well definitely no there are these are deep metaphysical assumptions in Indian philosophy which are said to have shaped the way actions in societies take place. Now with these classifications of karmas that we see as we see on the projection plate right now Prarabdha is one action that is accumulated look at it like a bank account. Now Prarabdha is some money that you have already worked for saved and accumulated in your account and you are beginning to spend it whereas Anarabdha karma in that way is that well these are stored money which is not yet been spent but which will be spent. Sanchit is that you have just earned it and it has reached your account and Sanchayaman is that you are in the process of earning it. So well while you have the privilege or the freedom of spending the money as you wish to spend in the case of karma you do not have that privilege. Now there are some questions which do concern this theory of karma that let me briefly put forth the questions right now that well number one is what about free will. Now if everything is depends on your past actions then where is the scope of free will that where do we find that human actions can be free because if every effect that comes forth is from a prior cause where can we have the novelty in this gap between the cause and effect. That is an easy question to answer for a karma theorist but let me list down the questions first that well let us say what about free will and second let me talk about well what happens to the self realized. Now this is a question that we have frequently talked about that we might have heard that well the realized soul destroys once destroys their own karmic accumulate and goes ahead but now the law of karma would perhaps not have it so that well karma cannot be destroyed let us so is there a conflict between the claim in Indian philosophy that well law of karma that karma continues and on the other hand that karma can destroyed with knowledge we will talk about that too. If you look at the third question do all actions have a karmic effect well the fact that I am writing in red instead of blue does this have a karmic effect well again this is a simple question to tackle that is basically tackling the difference between moral actions and immoral actions. Now an interesting question that does the law of karma I abbreviated as L ok for short limit the autonomy of God. Now this is a very interesting question I would like to bring you forth that well many of us have conceived Indian philosophy as very much theistic with an intervention of God. Now is law of karma making this notion of God impersonal that is well what there have been many theistic bakti seers and proponents who have claimed that well what use was this God if this God cannot forgive cannot do favors to us. Now law of karma in that way does not entail us to favors from God well if you have done something you will get the result of that act no matter what no matter how much you how much a benevolent God would like you to like to forgive you or to reward you. So the discretionary powers so to say of any God like entity is taken away by law of karma. So does this limit so is this law of karma an alternative to God does it critique or limit the autonomy of God or does it can the both go together both being very significant portions of Indian foundational metaphysical thinking. Then we have what is the beginning of this karmic chain. Now if everything is caused by something prior where is the beginning and sixth and final question that perhaps many of us would be thinking is that well benevolence and compassion towards the suffering would be intervening in the law of karma what do I mean by this well I mean by this is that well let us say if theory of karma is claiming that well that there can be an intervention in this karmic desert that we get let us say if somebody is suffering and in fact does the law of karma lead to fatalism because that is the crucial question that well if somebody is suffering the karmic theory would say that well that the suffering is a desert or an result of the earlier actions. And now as a third party or as an observer if I intervene to restrict or to alleviate the suffering exhibiting compassion or benevolence am I interfering with the law of karma am I interfering or intervening in the play of the law of karma this gives me an interesting analogy that well wildlife photographers analogically speaking when they enter or when they are permitted to photograph or video record or document wildlife and say natural reserves or forests well one strong standing instruction with them left for them is that they will in at no point ever intervene in the functioning of nature. So, well if they see a hapless bird which has fallen down from its nest and a predator being nearby the photographer or the documenter has been denied to get put the bird back into the nest or prevented from being the food of the predator. Now this is an analogical example of benevolence or karma or anything that interferes with the notion of karma that well benevolent action in fact this is brought forward to a very potent was brought to light by a very potent debate taken place in our pre-independence era when Mahatma Gandhi declared that the earthquake in Bihar which caused a lot of suffering as being the result the karmic result of the practice of untouchability to which Rabindranath Tagore has very aggressively responded that well this is just a factually inaccurate justification that surely the point that Gandhi would want to say is not that we should not help those suffering but that the suffering is perhaps caused by the karmic influence of practicing false policies or practicing unfair policies. Now this is the debate this is the question that we come when we talk about the sixth question that benevolence and compassion as towards the suffering would be intervening in the law of karma.


The reason that HIV was invented as the cause of AIDS is it was a way to get research money. That's how you get big bucks. That's how you get research grants and fund your laboratories and your research. In other words, if AIDS is caused by behavior, then there's no money in that because you just tell people, hey stop doing the behavior. But so that's why they have to find some bug that they can blame it on.  We need billions of dollars of research to kill this thing. So we're chasing after something, Dr. Duesburg, even if we got it neutralized that it wouldn't do a single solitary thing to stop the AIDS epidemic. Now last question. I got to get to this question before we're done. What does cause what we know is AIDS? What I know is AIDS, whatever they call it, what is abnormal diseases like Pneumocystis pneumonia and diarrhea and dementia in 20 to 30 year old men and women in the US, in Europe, in Asia, and even the few that fit that definition in Africa is recreational drugs like cocaine, heroin, amphetamines in the US. The CDC even reports that intravenous drug use is half of all AIDS patients.   And they can only accomplish that like our Olympians break records now in the Olympics by taking counts of drugs.


YouTube team keep it clean. What's going on is engraving here with another video and in this video We are here to review yet another list and y'all know me I love going over other people's lists the top five this the top 10 that the top 32 this Especially in this instant up, but I hate making my own. I hate it with a passion But in today's video, we're going to go over PFF who a lot of Ravens fans. They just like PFF. Oh, I Don't like that place But we are here to go over PFF's top 32 best outside corners not best corp best outside corners So that's important as we move forward through this list and let's just read the criteria Says NFL defenses will ask very different things of their cornerbacks depending on the scheme that they run Some cornerbacks are tasked with shutting down their side of the field and zone while other top cornerbacks are given The unenviable job of following the opposing teams number one wide receiver in man coverage It's part of what makes identifying the league's best at the position. So challenging So here is PFF's take on the top 32 outside Corners entering the 2021 NFL season featuring stiff competition for the top spot among several players under the age of 27 so very young guys so number one to get right into it Jair Alexander from the Packers That guy is amazing He is amazing him being at the number one spot Okay, no problem there at all whatsoever. He deserves it number two Jalen Ramsey Another one a beast Jalen Ramsey somebody that runs his mouth. He talks that talk But he backs it all up and he's somebody at the Ravens could have had but they didn't want to give up that additional pick Anyway, we don't need to get into that Jalen Ramsey continues to do what he does. He is a monster Number three. Oh Marlin Humphrey made it to number three. Oh, whoa, he is high up on this list Marlin Humphrey. Wow. Okay. Let's read in detail what they had to say about Marlin Humphrey We consider putting Humphrey a top the list of the best slot cornerbacks in the NFL Given that he has spent the majority of his snaps lined up in the slot over the past two seasons feeling the void left by an injured Tavon young Humphrey seamless inside outside versatility is a big reason why he ranked so highly here They're not done. The 50th cornerback out of Alabama has recorded at least 800 coverage snaps in the slot and out wide throughout his NFL career He also ranks in at least the 87th percentile in coverage grades at both alignments The most impressive part is that it all comes in a man-heavy Baltimore defense few cornerbacks in the league possessed that kind of ability As soon as I saw him at number three the first thing I thought of was wait a minute This is best outside corner than Marlin Humphrey. They just have him in a slot like a lot But I really appreciated that they put that in there the fact that he Has taken the place of Tavon young because you know a lot of times and it's to no fault of their own But a lot of times these lists creators and makers and whatnot They will just look at numbers. They'll look at numbers look at grades Look at this and that but they won't know the background of it But when you know the background of something it helps that much more so I can really appreciate that breakdown Oh, they got Marlin Humphrey number three. Oh, that's high That's good, but the versatility But again, it's talking about outside cornerback with Marlin Humphrey is on the outside Oh, yeah, he does this thing there. He does this thing there for sure I think Had he been able to really stay on the outside as a cornerback Which I think he'll get back to this season especially with the addition of Sean Wade and Tavon young coming back and hopefully Tavon young stays healthy But Sean waiters there's a Justin case guy But I think Marlin Humphrey can get back to being that outside corner that we know him to be instead of being in the slot Even though he did a thing from the slot Even though he's not a slot corner. He's an outside guy. So shout out to home. Anyway Number four Xavier and Howard for the Dolphins. This guy gets like 25 picks every single year, man He gets like 25 picks a year. It's crazy You would think that he was like a safety or something and it's weird because he keeps getting picked But these these quarterbacks they keep throwing at them. They keep throwing at them See now, that would be a backstory that I would like to know And you know what? Let's let's read it I said Howard is but one of the best making play one of the best playmaking cornerbacks that the NFL has to offer since entering the League in 2016 18.7 of his targets and coverage have resulted in a past breakup or an interception Leading all cornerbacks who have been targeted at least 150 times over those five years How was 2020 for performance put that playmaking ability on full display with double digits past breakups and Interceptions to go along with an eighty nine point six coverage grade Okay, well, thank you BFF for the breakdown number six James Bradbury from the Giants Okay, a little sleeper right here James Bradbury. I think he used then he used to play for the Panthers I want to say he did I think he did But anyway, he's number five number six Currently as of the recording of this video on June 3rd current Patriots cornerback Stefan Gilmore It's been a lot of buzz in the air about him possibly being traded nothing yet So it's like he on the same boat as Julio nothing yet So we'll wait and see But yeah, Stefan Gilmore is at number six really good cornerback Mmm, just imagine and I know it wouldn't happen, but just imagine The Ravens could have had them all in Humphrey had the Marcus Peters Jimmy Smith. They want you They asked if I'll get more to them. I just be a man. I just be thinking of that stuff sometimes Anyway, number seven Tredavious white If we're thinking about it, I would probably put him At number three or four like straight up man, I'll probably put him in number three or four Tredavious white, I will definitely put him top five For sure But so I think he's a little bit low even though he's number seven. I still think he's low number eight Denzel Ward Denzel Ward is nice nice cornerback But again the health concerns. That's his biggest knock on Denzel Ward nice corner very good cornerback, but the health concerns That's tough, but very good when he's on the field excellent Number nine William Jackson the third former Cincinnati bingo belief Went to the Washington football team. Let me read about him cuz I'm not I know and he used to play for the Bengals But and I know he's very very fast, but I know he was like that like that Well, he's at number nine said Washington possessed one of the better defenses in NFL in 2020 Jackson's edition should only improve it giving them a number one cornerback will allow them to play more man coverage If they desire Jackson's burst onto the scene with a 90.4 coverage grade in its first season of action back in 2017 he allowed just 30 receiving yards on 359 coverage snaps across his final 11 games that year Jackson hasn't quite reached those lofty heights in the three years since But he is coming off his best statistical season in 2000 since 2017 with just 52% of the passes into his coverage being completed in 2020 I'm confused because they haven't met number nine, but the way that they worded that whole thing It doesn't make him sound like he would be at number nine, but anyway Number ten. Oh, two Ravens in the top ten. I guess Peters and Pete juice me Now we know about the playmaking ability. We know about it and we love it and we appreciate it I've liked Marcus Peters even before he came to the Baltimore Ravens that dude boy that boy got some attitude on him and Perfect fit for the Ravens because it brings life. It brings energy Sometimes attitude can get the best of them and things going bad It can be a big yikes, but when things going good, it could be a big yikes for the opposing receiver And again with Marcus Peters when he's lined up against you and he gets to clapping like that Yeah, that means you have been having a bad game and he has been having an amazing day But let's read what they said about Marcus Peters Peters and Humphrey are the only top 10 duo On this list and a big reason why Baltimore secondary has had the success that it has had over the past two seasons It's impossible to discuss what Peters brings to the table and not start with his neck for getting his hands on the football as his 31 interceptions since entering the league in 2015 are nine more than any other cornerback. That's crazy That's crazy Anyway, and Peters ranks tied for fifth at the position and pass breakups over that same time frame Okay, so Peters that he be getting his hands on the ball more ways than one not just picks for the past breakups That's something that I didn't realize but I appreciate a lot his aggression will burn him on occasion Oh, but you can't say that Peters doesn't make up for it with the positive He brings to the table say that's everything everything that I said to man everything that I said they yeah, they nailed it They nailed it. So thank you PFF. We appreciate that next up at number 11 Richard Sherman Okay, I want to read about this one Richard Sherman says that and he's a free agent as of this video June 3rd There's a perception that Sherman's better days are behind them But entering his age 33 season There's still plenty of room for Sherman to take a step back and still be one of the better cornerbacks in NFL He and the rail readers are the only two cornerbacks with at least 250 targets since to 2010 Who have allowed fewer than 50% of those passes to be completed? Injuries held them to just over 300 snaps for the 49ers last season, but Sherman was the highest graded cornerback in the league back in 2019 He's one of the more impactful free agents still on the market So now it's like Okay, now you're starting to lose me a little bit now This list is starting to get a little bit shaky now But let's keep going again. It's the top 32 outside cornerbacks heading into this season number 12 adori jackson On who's on the giants former tennessee titans first round pick adori jackson All right, let's get a little real shaky now number 13 steven nelson another free agent He used to play for the chiefs. He played for the Steelers and now he hasn't been picked up yet I remember with him, um, right before he got released Uh, he tweeted something Like showing that he wanted to get released and then boom not even an hour later. He was out of there Uh, next Um 14 byron jones from the dolphins. Okay. So dolphins got two cornerbacks in the top 15 Byron jones was their big free agent acquisition from uh, the dallas cowboys And he was known not for getting an exception but for not letting that receiver catch the ball He was known for being locked down and locked up but not for getting picks, which is fine That's still you still want that in the cornerback cornerbacks number one job is to not let the receiver catch the ball That's a number one job So he did a good job of that. So shout out to byron jones Uh, number 15 carlton davis. Oh, wow okay So in number 16 jc jackson. Oh, that's that's mr.  Oh Joe hayden said hayden may be on a downward slope of his career But he has provided Pittsburgh with steady play at the cornerback position since joining the team in 2018 Yeah, wow, he's been there for a little while now, man That's crazy. It time flies man. Anyway number 18 march on ladder more oof Now march on ladder more if you put all his games against mike evans, he will be number one If you just if you only graded it by his games against mike evans, he'll be number one He locks that dude up every time. Uh, number 19 jamil dean Hmm. I'd have no knowledge about jamil dean at all At all i'm very naive to jamil dean number 20 Darius williams. Oh from the rams former baltimore raven Darius williams He a it worked out for him. So shout out to him for making it Number 21 kyle fuller from the den for broncos number 22 jason forrette from the 49ers now He used to be with the chargers. I believe too Uh, number 23. Oh big play slay there is slay from the eagles Their big acquisition from last season. So shout out to big play slay number 24 Bradley roby from the texans. I think he started off his career with the broncos. I believe not 100% sure what I believe So anyway, 25 she killed griffin. Oh, yeah jack was I was about to say seahawks But he left there and his brother's getting ready to His brother's a free agent now that his brother's a pastor outside linebacker So he's getting ready to be on a new team too because I don't think seahawk signed him back Number 26 kinder fuller from the washington football team number 27 genorge jinkins from the titans 28 malcom butler from the card Malcolm butler that guy Number 29 is ronald darby from the broncos number 30 Xavier rose from the colts. Oh, okay Xavier rose man when he was with the vikings, he was like up and down And then the vikings I guess they were just like, okay, bye. You into the colts solid 31 tray wanes Now tray wane. I know he was always known for his speed, but I thought he had a it was a big yikes in his cover skills But according to pff number 31 and last but certainly not least is uh kasey hayward jr From the los vegas raiders who ravens will be getting an up close and personal look at Come week one of this season on monday night football So shout out to pff. Uh, this was a fun list as they always are I I always enjoy these pff lists whether I agree or disagree with a particular player or the list as a whole I always enjoy them and I appreciate them for making these lists because they are very uh fun to go over um and just to really Just to really share with everybody too and to give my thoughts on it and to hear everybody else thoughts on it too So y'all let me know what you think about this list. Who was wrong? Who was right? Who should have been better? Who should have been worse? And on that note we out


A lot of Republicans, conservatives, they continue to accuse Democrats and progressives of being socialists. Socialist shopping lists. Radical socialist views. They want to ram through their socialist vision for this country. But in fact, the irony is that corporate welfare is a form of socialism. It's socialism for the rich. Corporate welfare is a lot of subsidies and tax breaks and all sorts of special privileges going to corporations. Large corporations in the United States are now paying a fraction of the taxes that they were paying 10, 20, 30, 40, 50 years ago. And why is that? Because they've got the lobbyists and because they have a tremendous infrastructure of influencers.   It's a form of corruption of our economic system. Corporate welfare costs American taxpayers hundreds of billions of dollars a year. The biggest corporate welfare in America is where? It's on Wall Street in terms of the too big to fail banks. It's in the pharmaceutical industry in terms of the huge amounts of basic research and development that taxpayers pay for. High tech, you see the same thing. The Department of Defense just decided that Amazon would get the gigantic contract for cloud computing for the Department of Defense or take the war in Afghanistan and just look at military contractors like Boeing. Boeing spent about $16.7 billion on lobbying in 2017 and won a $23.4 billion contract from the Defense Department. Oh, and their tax rate was only about 8.4%. Isn't that corporate welfare? We are providing all sorts of benefits for people at the very top, CEOs, major investors. The typical CEO today is making 320 times what the typical worker is making. We haven't seen that ratio, that extreme ratio since the 1920s.  The public is paying for this. They have even convinced a big portion of the public that you don't want to tax corporations because that would, what, stymie innovation, that would stymie the economy? Well, that's baloney. It also keeps big companies fat and happy, makes it difficult for smaller competitors to get in. It reduces the number of startups in terms of the economy because these big players get so many benefits that the small competitors really don't even have a chance. Corporate welfare clearly is a form of socialism for the rich. These people at the top are not responsible. There's no accountability. This is the worst form of socialism.


Germany's Rheinmetall to deliver Lunar Drone System to Ukraine. Rheinmetall, Germany's automotive and arms manufacturer, will deliver a Lunar New Generation Drone System to Ukraine by the end of 2023, Bild am Sonpag, media outlets reported this, citing company sources. The package will consist of a ground-controlled station with several drones, a launch catapult and military trucks, according to Bild. The drone system could be used as a reconnaissance system, provide an LTE network, and intercept or jam communications. Drones are integrated into every phase of fighting on both sides, Russian and Ukrainian, with extensive fleets, air defences and jamming systems. A report from the UK-based Conflict Armament Research published in early August, indicated that Russia has started making copies of attack drones obtained from Iran in the previous year and has been using them against Ukraine. Iranian-made Shahead Kamikaze drones have been one of Russia's most effective weapons. Rheinmetall was identified as the previously unknown buyer who recently purchased 50 Leopard 1 tanks for Ukraine from the Belgian company OIP Land Systems, the Handelsblatt newspaper reported earlier. The company purchased the tanks, which will be refitted in Germany as part of its future exports of armored vehicles to Kyiv.