Now we will solve a example problem about this design of counter fort retaining wall. Now look at this example, it is given design a counter fort retaining wall for this following given particulars. First one is given height of wall above ground level which is equal to 5.5 meter, then safe bearing capacity of soil safe bearing capacity of soil is given 160 kilo Newton per meter square, phi value is given phi is equal to 30 degree and unit weight of soil is given 16 kilo Newton per meter cube spacing of counter fort spacing of counter fort counter fort about three meter three meter and R C C unit weight unit weight of R C C or unit weight of R C C which is equal to 25 kilo Newton per meter cube and use M 15 grade concrete and F e 250 steel. As I said earlier now this is a typical example problem we will have to design for counter fort retaining wall. No dimensions has been given dimensions has to be taken derived then your stability analysis then your structural design it is given height of wall above ground level is 5.5 meter that means from ground level to above is 5.5 meter and the soil below the ground level it is bearing capacity is given 160 kilo Newton per meter square phi and gamma of the retaining soil retaining soil is given and spacing of counter fort walls is about three meter and unit weight of R C C reinforced concrete 25 kilo Newton per meter cube and it has been the plan to use M 15 grade of concrete and 250 steel mile steel 250 for designing this counter fort retaining wall. Now first step is to find it out this wall has to be if I take this retaining wall like this if I go for this counter fort retaining wall like this what will happen you will have to consider some depth as a depth of the foundation as a depth of the foundations. This depth of the foundation generally provided about 1 to 2 meter either you can assume or you can this depth of foundation you can find it out from this depth of foundation you can get it p by w into 1 minus sin phi by 1 plus sin phi whole square where is your p is your say bearing capacity of soil pressure in this case and it is coming about 160 thousand divided by your w is your unit weight of soil based on that it is coming about 16 thousand into one third whole square which is coming about 1.11 meter. So, let us take about foundation depth is equal to 1.2 meter foundation depth then once you are taking this foundation depth is equal to 1.5 meter then overall height below the above the ground surface overall height of wall overall height of wall overall height of the wall is equal to 5.5 plus 1.2 which is equal to 6.7 meter if you look at here it is given height of wall above ground level the ground level is here because retaining wall cannot stand above the ground level you have to make sure that there is some foundation depth retaining wall has to be put it below the ground level. So, this height is given about 5.5. So, this depth of the foundation is we have taken 1.2. So, overall height of wall is coming about 6.7 meter now thickness of the base of the wall now the dimension one this overall height and the depth of the foundation is over part one is your dimension of the retaining wall dimensions of the retaining wall in this dimension of the retaining wall first find it out your thickness of base slab thickness of base slab you can get it as I have said earlier for counter forward retaining wall 41.7 root over of h it should not be lower than that and 20 l root over of h and it is coming about to be 1.5 meter. 324 mm and this coming about to be 402 mm. So, this thickness of base slab should not be lower than this. So, now we can take a higher value let us provide thickness of the base slab say 400 mm 400 mm. So, now the actual height of upright slab upright slab is your this is your upright slab this is your applied slab. So, thickness of base slab we have found out it is to be 400 mm that means 0.4 meter. Now this upright slab for upright slab actual height height of upright slab height of upright slab is 0.4 meter. Is equal to 6.7 minus 0.4 which is equal to 6.3 meter 3 meter. So, this height is about coming 6.3 meter. Now depth is coming about depth of the foundation it is coming about is equal to 1.2 meter. Now we will have to find it out this particularly design for your upright slabs in case of upright slab. What is your pressure intensity in upright slab for upright slab what is your pressure intensity. Pressure intensity is equal to gamma k a h which is coming about 16000 into 6.3 into 1.3 this is coming about 33000 into 1.3 600 kilo Newton per meter square. Now maximum bending moment is coming about to be maximum bending moment which is equal to W L square by 12 or P L square by 12 P is your pressure intensity P is equal to your pressure intensity maximum bending moment is equal to P L square by 12 from there we can find it out P is equal to 336000 into 3 square by 12 and this comes out to be 25200 Newton meter. So, once you get maximum bending moment then what you are what we can suppose to get it we can find it out your area of steel this mild steel for m 15 grade concrete the permissible stress in concrete c is equal to as I said earlier it is equal to 5 Newton per m m square for for for m 15 grade concrete. Then for mild steel F e 250 steel permissible stress in steel t is equal to 140 Newton per m m square based on this if I equate maximum bending moment to area of steel this should be come out to be you can either equate maximum bending moment to area of steel or maximum bending moment to your q b d square because you have to find it out this thickness also we got it base slab thickness now we will have to find it out thickness of your operate slab at the base. So, from this q b d square which is equal to maximum bending moment. So, q for m 15 and 250 steel it is 0.87 b into 1000 into d square which is equal to 252 into 100 into 1000. So, d is equal to 170 m m now if I am using effective cover of 40 m m this is your effective cover then it comes out to be 210 m m now let us say thickness of operate slab let us say 210 m m 210 m m you can keep this thickness of this operate slab either you can keep it this way or you can keep it continuous throughout from top to bottom this thickness will be same there is no varying in thickness. So, in this case we are considering this thickness uniformly throughout that means uniform throughout this thickness we are considering this this about 210 m m then after getting the thickness means width of your operate slab at the base it comes out to be 210 m m then thickness of your base slab is 0.4 meter or 400 m m. Now base width this base width has to be come out and this width has to be come out then we will go for your stability analysis next part is your base width. So, base width is generally varying from 0.6 h to 0.7 h. So, if I take total height h is equal to 5.5 it is this height operate slab is your this height is there. So, if I am taking into this is coming about to be 6.7 into 0.6 6.7 into 0.6 and this is your 6.7 into 0.7 it comes out to be 4.02 and 4.69 meter let us provide let us provide this base width is 4.5 meter. So, base width is comes out to be 4.5 meter. If you look at here this base weight base width is generally 0.6 to 0.7 times of total height of this retaining wall. So, total height if I am taking it 6.7 5.5 plus 1.2 this is coming about to be 6.7 6.7 into 0.6 6.7 into 0.7 this base width you can take it between 4.0 to 2 4.6 nanometer either you can take it 4.03 4.3 4.5 4.6 just I put it any value between this 4.5 meter base width. Now next part is your toe projection toe projection means how much it will be projected if this is my retaining wall and this is your toe and this is heel and this part is called heel slab heel slab and this is your upright this is your upright slab this part is your this entire part moving towards this this is upright slab. Now how much is your toe has to projected beyond this retaining of soil mass. So, this toe projection comes out to be it says that one fourth one fourth of b one fourth of b and this value is your b. So, this comes out to be 1.1 meter. So, you can provide either 1.1 or you can provide a toe projection of 1 meter. Let us provide this toe projection of 1 meter toe projection of 1 meter. Now if I provide this toe projection of 1 meter that means this dimension is almost over tentative dimension is almost over. Let us make a sketch how the dimensions of this it looks completely retaining wall you can show in a sketch of your tentative dimension for your stability analysis 1.1 meter and this is about to be this is about your 1 meter and this is your 210 mm and this is your 0.4 meter and this is your 0.4 meter and this is your 3.29 meter. So, with this with this this is your tentative dimensions and now with this tentative dimensions will check this stability analysis if this stability analysis satisfies four factor of safety one is your factor of safety against overturning other is your factor of safety against sliding and e should be your eccentricity should be less than equal to b by 6 and bearing capacity that means the dimensions are ok. If this stability analysis if it is not satisfy either you modify the dimension or you can go for some remedy. So, we can check this now this stability analysis calculation of this forces we can check how it is how it is coming about means total resisting force and driving forces all has to be calculated then once you calculate then you can go for your bending movement and pressure at the base all these things. Now if I name it this is w 2 this is w 1 w 1 w 2 you can name it w 3 also this part is your 6.3 this part is your w 3 now start 1 by 1. So, w 1 is your what is your w 1 w 1 is your weight of this upright slab from here to here. So, that means unit weight what is your width 0.21 into 6.3 6.3 6.3 this to this height is your 6.3 into unit weight of concrete is 25000 which comes out to be which comes out to be 33075 kilo newton. So, this is a vertical force because this is vertical downward by self weight this is vertical downward similarly you can find it out w 2 w 2 is your weight of base slab w 2 is your weight of base slab which is about to this is this width is your 4.5 meter this width is your 4.5 meter. So, w 2 is equal to 4.5 into 0.4 into 25000 which is coming about to be 45000 and this is again vertical if I am putting it this symbols and this symbols this means force is acting vertical this means force is acting lateral in that direction. Then third part is your w 3 w 3 is equal to 3.29 into 6.3 into 16000. If you look at here 3.29 is this width and this height is your 6.3 with this counter 4 soil has to retain. So, unit weight of soil is again this is vertical 16000 and which is coming about to be 3 3 1 6 3 2 again this is vertical. Then fourth one is your earth pressure earth pressure that is your k a gamma h square by 2 which is equal to one third 16000 into 6.7 whole square by 2 then. So, earth pressure is there then moment you can find it out from this moment this is your pressure. Then moment you can find it out k a gamma h cube by 6 which is equal to one third into 16000 into 6.7 whole cube by 6 now this is the vertical force and this is your lateral forces or horizontal forces acted by this directions. So, these are about your force now you can find it out lever arm lever arm lever means means this force acted how far from toe or how far from your heel. So, in this case from this toe this w 1 is coming about to be 1.105. So, because it is 1 meter and 0.21 by 2. So, this is coming about 1.105 and w 2 is your it is straight forward is half it is coming about your 2.25 and w 3 is coming about 2.5. So, 855 and earth pressure will act at a distance 6.7 by 3.  If you look at this how this clockwise moment all vertical forces this will give a clockwise moment lateral forces earth pressure this will give your anti clockwise moment. That means if I write is clockwise moment and anti clockwise anti clockwise moment and from this you can find it out this clockwise moment all the term for w 1 it is coming about 36547.8 and this is coming about 1.01250.0 and w 3 is coming about 946809.36. So, and anti clockwise moment is coming about 267345. So, this comes out to be 108467.2. So, now net moment you can find it out net moment is equal to clockwise moment minus anti clockwise moment which is coming about to be 817262 0.24 Newton meter. Now we can check you can find it out eccentricity you can find it out this is the resultant force are acted. So, how far from your c g how far from your c g you can find it out e. So, first find it out your x bar distance of your eccentricity x bar from the toe which is equal to m by v total moment resisting as well as clockwise as well as anti clockwise and total vertical forces it is coming about to be 817262.24 by 4097.07 which is equal to 1.99 and e is equal to e is equal to b by 2 minus x bar which is equal to 0.26. So, b by 6 is equal to 0.75 that means e is equal to less than b by 6 which means there is no tension crack. One stability criteria has been satisfied we find it out what is your eccentricity eccentricity should be less than b by 6 that means there is no tension crack that means there is no tension crack below the base of the slab there is what do you mean by tension crack once there is a tension crack there is a gap between soil and the wall. So, there is no gap between soil and wall. So, similarly factor of safety against sliding you can take from this factor of safety against sliding it is coming about 4.0 which is I just leave it for you then sorry factor of safety against sliding is coming about 1.711 you can assume mu is equal to friction between concrete and soil is equal to 0.5 you can assume this friction between concrete and soil is equal to 0.5. So, 1.771 now which is ok which is greater than greater than your 1.5. So, and your factor of safety against bearing capacity also it is ok it satisfy now we can find it out what is the value of p maximum at the base and p minimum it will comes it will be calculated with v by b 1 plus 6 e by b and it will be calculated v by b total vertical force v by b 1 minus 6 e by b and this comes out to be 1 to 1 394 Newton per m m square and this is equal to 0.5. This comes out to be 6 0 6 9 7 Newton per m m square. So, now if I draw it if I draw it this is my pressure distribution is coming. So, this pressure distribution this part is coming about 1, 2, 1, 3, 9, 4 then 6 0 6 e by b 1 minus 6 e by b 1 minus 6 e by b 9 7 this is your pressure distribution around this below the base of the slab. What I did after the after the dimension has been fixed based on the calculation or based on the assumption value then you can do it systematically by taking into your forces which your vertical forces lateral forces then clockwise movement and anticlockwise movement. So, I have made it into very small small part. So, that there will not be any mistake I make it upright slab as w 1 base slab is your w 2 soil retaining in the counter for part this is your w 3 then is your how much is the earth pressure coming all I have taken into small small part and total movement has been calculated. Once you calculate your total movement and total vertical force also you can calculate from here from this these these these what is your total vertical forces. So, x bar you can find it out from there you can find it out eccentricity and you check it whether e is less than b by 6 or not if e is less than b by 6 that means there is no tension crack 1 factor of safety satisfied. Second factor of safety again sliding has been checked I leave it for the student to calculate it is about 1.711 which is greater than 1.5 mu has been assumed as 0.5. So, and also factor of safety against overturning as well as factor of safety against bearing capacity has also been checked it is within this limit then once this stability analysis has been done then what will happen this dimensions whatever we have taken that is ok now that is ok now. So, step one dimension is ok step two stability analysis has been done step three is your design structural design of your RCC counter for retaining walls. So, from this stability analysis we calculated P maximum maximum pressure below the base of the wall minimum pressure below the base of the wall and now in the next class we will go for your structural design of your counter for retaining wall.


Cassini's Grand Finale Presented by Science at NASA NASA's Cassini mission will come to a spectacular close on September 15, 2017 at 5.07 a.m. Pacific Daylight Time after almost two decades in space since its launch in 1997. The spacecraft has been orbiting Saturn and studying that system for 13 years. It's now checking off items from a daring bucket list as it makes the most of its final months. Since its arrival at Saturn in 2004, Cassini's stunning observations of Saturn, its rings and moons, and the surrounding space have produced a series of extraordinary discoveries. Cassini confirmed the presence of huge lakes on the moon Titan's surface containing liquid methane and ethane. This makes Titan the only body in the solar system other than Earth that has bodies of standing liquid on the surface. Cassini also caught the moon Enceladus spewing plumes of water vapor, icy particles, and organic compounds through fissures in its frozen outer shell known as tiger stripes. In addition, the mission showed that Saturn's rings are an active and dynamic system that can serve as a laboratory for scientists to study how planets form. The spacecraft has finished a series of 20 pole crossing orbits between a gas giant's outer F and G rings, making new discoveries about the rings and the moons that orbit in the outer rings. Cassini is now in the process of executing 22 daring grand finale dives that are bringing it closer to Saturn than any spacecraft has gone before, in the 1,200 mile gap between Saturn and its innermost ring. Initial orbits have already produced a surprising discovery that is relatively empty between the planet and its rings. The images and data gathered during these maneuvers will add to the knowledge Cassini has already provided about how giant planets and planet families throughout our universe form and evolve. Researchers will be able to map Saturn's gravity and magnetic fields near the planet in detail, revealing new clues about the planet's internal structure. They believe that it is not as simple as a core of ice and rock with a huge envelope of hydrogen and helium on top. These final data will also greatly enhance what Cassini has already shown us about the age and origin of Saturn's rings and how much dust and ice they consist of. The conclusion of the grand finale comes when Cassini performs an epic final plunge into the gas giant's upper atmosphere. The spacecraft will transform into a fiery streak in Saturn's skies, removing the chance of a contaminating collision with any of Saturn's icy moons, and bringing a long, rich mission to a dramatic close. For more news about Cassini and Saturn, stay tuned to science.nasa.gov.


Hey everybody, thanks for tuning in to the second episode of Riddle Red Riding Hood Saga. Hope you guys enjoy! I wonder where these flowers are? Oops! Mr. Cow, why would you do that? Hehehe! Oh, there they are! Wow, these are beautiful! Let me put this basket down so I can pick some flowers real quick. These ones look good. They're not too, too dead. Perfect! Now that I got a nice bouquet, let's get the grandmas! What a nice wolf to recommend these flowers! I don't meet many wolves, but that one sure was nice! Man, oh wow, it's getting late! I gotta get there quick! Let's hurry! I'm in the room! Hi grandma! Hi Little Red Riding Hood! Wow, what's wrong with your voice? I'm very sick. Why don't you come closer? Wow, your ears are huge! All the better to hear you with. Look at your eyes! They're so big today! All the better to see you with. And your hands! They're so big! All the better to caress you. They're also so furry! That's to keep you warm when you're in my arms.  All the better to kiss you with. And your teeth! They're ginormous! All the better to eat you with. Get over here!  Wait, what is that? It kind of sounded like it came from grandma's house. I'm about to go check. Hopefully she's okay. Because those screams sounded kind of loud. I have to get up these stairs. I have to get up these stairs. Let me hurry. Grandma? Help me! Red, red, red, move! Red, move, move! Oh, what is that? I'm gonna have to take my gun out right now! Take this! Is that? He's gone! Red, are you okay? Thank you so much! Red, are you okay? You saved my life! I was so scared when I heard those screams. Like, hopefully you're okay. I was gonna be okay, Red. She's dead! Thanks, everybody, for watching the second episode and the last episode of The Adventures of Little Red Riding Hood. Thanks for watching. Hope you guys enjoyed. Remember to like and subscribe below.


Alright, so to start this project, you're going to need a few items. You're going to need a pair of scissors, a mat for your Cricut machine, heat press, tape, some sublimation paper. I use the A sub sublimation paper and then you're going to need your freshener parts. So I will show you what this is. I bought this. It came in a package off 150. So you get the string like here.  I looked up several of these and this one was the thicker one and I wanted a little bit of a thickness to this. So I will show you what this looks like. So it looks like this pretty plain, but it's got a nice thickness to it. It's not going to break. It's not very flimsy. It's nice and thick. So that's what you're going to need and you're going to need a sublimation printer as well. I will leave all my supplies that I use down in the description below. So let's get started. Alright, so I'm going to show you start to finish how to do these. I'm starting in design space. You can use this in Canva as well, but I'm so used to design space. I'm going to show you it this way. So I'm starting with a blank project here. I'm going to go to upload and I wanted to show you if you're not familiar with getting stuff from design bundles into design space. So I'm going to do that first part here. If you are familiar with it, please feel free to skip ahead. So I purchased these from design space. I'm going to show you really quick. I purchased a whole bunch of these and I thought they were really cool. You can see they were like $2 apiece. So they really were not that expensive. So with this one right here, I wanted to do this one, alcohol ink air freshener. So I thought that looked really cool. So all you got to do is click download files, click on your little download button there and yours might be different than mine. So it automatically shows me right here in my download file. If you don't see this, you have to find on your computer where it says download. So all I'm going to do is double click that. It's going to open up my download file. Now I can get rid of this and show you guys the next part. So as you see here, it has it right here and it's in a zip file. So what you want to do is double click on that and this is all the images that you just purchased. Now I'm going to shrink this just a little bit so we can see these both together makes a little bit easier. So I'm just going to take this and find them. So I'm going to show you really quick what they look like. So this is what this first one looks like. It's beautiful, but it's a circle. I don't want a circle. So I'm going to find the one that's just a square. So I believe it's this one, the R 2.7. So we're going to look at it really quick and I'll show you what that one looks like. So this one turned out to be a square, like I said. So I'm going to use these ones. So every one I'm going to be using is going to say the R 2.7. The other one I believe is just a lot bigger than the last one I showed you.   I'm going to go ahead and click complex and then click continue. Then if you wanted to get rid of any parts in the background, which I do not, I'm going to leave it the way it looks. If you did, you want to go ahead and click your race and do it that way. But these files are already come finished for you. So all I'm doing is uploading them into design space. So I'm going to click apply and then I'm going to always get this one, print and cut. You can see this one, the cut image. If you were just cutting this image out, nothing would show up. You want to make sure that you click on the print then cut. Then you want to make an image name. So they name this one already, but I don't want it to be named that. So what I'm going to do is I'm just going to get rid of all of this and I'm going to leave it say freshener. And then what I'm going to do is I'm just going to copy this for everyone. So I'm going to just select it and I'm going to copy it for everyone. It's going to be named freshener. I'm going to upload it. And then that's how you do it. I'll show you one more time. So here it is here. Upload image. I'm going to want to go and find the one that says the 2.7. So here it is here. So I'm going to take that, throw it in there. Go to complex, continue. Don't have to do anything here. Applying continue. Make sure you click the print and then cut and then just change your image name. You can do it to whatever you want. If you want to do an air green freshener, whatever you're going to remember that it is and then click upload. So I'm going to go ahead and finish it. That's all I did with these other ones here. I just found the size that I wanted and I put them in that way. So I'm going to go ahead, fast forward this part so you guys don't have to see it. But I'm going to put every single one of them that I got in here so you can kind of look and see if you like these. All right. So now I have them all in here. You can see what they look like. I thought they were really beautiful. So I chose these couple of things with these. I'm just going to pick a few of these really quick and show you what they look like when I print them out. So I'm just going to grab a few of these ones here. And also, if I go into view all, I will show you. I got some other ones too that are really awesome that I already printed out. So I had these beautiful butterfly ones, but then I also got these really beautiful sunflower ones for my daughter that I thought she'd like. So I did those. So I'm just going to go ahead and I'll add that to canvas and then I'll click view and then I'll show you what they look like. I'm going to make my screen bigger so we can see what this looks like. So it'll bring all of them up for me. The only thing you're going to want to do with this is just change the size of it. So for whatever reason, it didn't save my other one. So I'm going to go back to upload and then just pick the other ones that I wanted. So we'll do just a couple of these. We'll pick one of these beautiful hummingbirds add to canvas. Sorry, I'm not sure what I pushed that I didn't make it come. So we'll show you what these all look like. It's going to take a minute because it's just a little bit bigger images on here. So we're going to go ahead and shrink this. We'll shrink this one just for the purposes of the video. So I will show you what these look like really quick. I absolutely love these watercolor ones. They look amazing. So right off the bat, this one. So it looks like a circle here. I'm going to go ahead and delete that one just because that extras you're not going to see that around the edge. So I'm going to go ahead and shrink these up. So for my particular image that I'm going to be using, this is the size that I'm going to be doing. So I'm doing 3.2 by 4.2. So I'm going to click on the first one, unlock it, and then we'll change it to be the 3.2 by 4.2. And I'm doing it a little bit bigger than my actual freshener. But I'll show you why when we go to the heat press machine. Pretty much the reason is I just want a little bit bigger than my freshener. But again, I will show you when we get to my heat press why I did that. So I'm just going to go ahead and go in here and change all of these to that size 3.2 by 4.2. But you want to make sure that it's the size that you want. So I will leave the link down below for my fresheners that I purchased and then you can kind of go off of there but test it just because I'm using a specific size doesn't mean it's going to be the same size for use. You always want to check that out. So I'm just going to go ahead and change all of these to the correct size and then I'll show you the next step. All right. So I think I got them all. I'm just going to go ahead and click on each one of these, making sure I got them all correct, which I did. OK, so basically the only thing that you have to do with this is click make it. So you're going to see a thing on the side here. It's giving me a little bit of a warning here. It says low resolution. So it's saying that just because it's going to be printing it. So when I used and I printed this already, they turned out absolutely gorgeous. So I'm going to show you what they actually look like. That's all you got to do. So if you wanted to make your own like I'm going to show you really quick. I'm going to click upload. I actually got a picture of my dog Ray and I printed one of him out. Oh my gosh, it is the cutest thing. So this is my puppy. His name is Ray. I have another dog that I have to do on his name is Baxter. So I wanted to make one of these for my daughter for her car because it's actually her puppy. So I did the same thing. I just resized it. You can use any photo that you want. So if you have a best friend, if you have a boyfriend, if you have a girlfriend, if you have friends pictures, if you have an animal picture, you can do whatever you want. You just upload it into design space and you just resize it, but you want to make sure the quality of the picture is good. So I'm going to delete that one since I already have one of those, but I will show you what that looks like. Other thing you want to do is make sure you just resize it. That's it. Click make it and let's have some fun and print these out. So on this side here, I don't know why I have to change my settings. It always goes to eight and a half by 14. My printer only goes to eight and a half by 11. So I have to change that. So just in case yours pops up like that. Make sure you're doing it the correct size. The other thing that you have to do with these, if you wanted to make these double-sided is to make sure you do two of these. So it says project copies. I'm going to do two because I want two of each one. So I want to I want two of these images and I want two of these images. So I hope that makes sense. And the other thing you have to do is mirror your image because we're going to be using heat press. So anytime you're using your heat press, you always want to make sure you mirror your image. So I'm going to go ahead send this to my printer. It's going to cut out two of these. So two exact ones. So I can put one on the front side and one on the back side. So I'm going to go ahead and click continue. I'm going to send it to my printer and I will let you guys know in the comments down below what printer I use. It is the absolute best. It was a regular Epson printer and I converted it into a sublimation printer. Very, very simple. Before you even use it, put sublimating ink into it. That's the only thing that you have to do. I think mine is a 2800 but I will double check and I will leave it in the supplies down below and I will leave all of my supplies down below for you to look at. So for mine, mine is actually this one. Oh, here we go. Epson equitank 2800. That's what I have. So I have two of them actually. I have a colored printer that I used a regular ink for and then I changed this one to a sublimated printer. So I have the equitank 2800. So I'm going to use that and then I'm going to go here where it says add bleed. I take that off. So here's the other spot you can do it. You can make copies. You can do two right here. So that's what actually I'm going to do. So I'm going to cut out two of these and then I'll do two of these as whoops, two of these as well. So I'm going to go ahead and click click print. It's going to print two of those for me, but then you also want to make sure you're printing your second image as well. So I want to go up to the first one that I just missed. Go to send to printer. You can see it right here. Maybe here my printer in the background. It's going change it again. Take it rid of bleed and then I'm going to make two of these and print it. So once that's done printing, you just want to take your mat, your regular green mat. I would suggest if you have a very sticky green mat, like a brand new mat, I would use a blue mat. You do not want very, very sticky. You just want it sticky enough that it's going to hold your paper down. Then I use this heavy cardstock right here. I went into browse all materials and I typed in cardstock and I found this heavy cardstock hundred pound. That works perfect for my paper. Another thing you need to know and again, I will leave everything in the description down below. You need subbing paper. Very, very important. You cannot just use regular paper. So I will leave all my supplies down below and I'll let you know which supplies that I use. So once those are done printing, I'm going to put it onto my Cricut machine. So I'm going to get started with that and then I will be back and I will show you how to use these at your heat press. All right. So here is my heat press. I have the HTV heat press. This thing is absolutely amazing. I love it. I've done a few videos on this radio. So if you're new to this, go back and check those out. But I absolutely love it. There's absolutely no pressure on here that you have to twist or turn or pull down or anything with this. You should slide it in and out. It's absolutely amazing. So I have it set at 385 degrees for 45 seconds. That is what works for me. Here are some of my images that I'm going to be using. I put them together so that I know that I would be using one on each side. But here's just a few of them that I have so far. And then this really pretty flower. So what I'm going to do is take one of my pieces here and I'll leave the link down below. You can see it's very thick. This isn't thin. This is very nice and thick. And before I remember when I told you that I would tell you why I made it a little bit bigger is because of this. So here's an image. I'm going to place this over top. But now I have a little bit of room on each side to play with that if I wanted a higher or lower or anything like that. Then I'm going to take a piece of heat tape right here. I'm going to put this upside down. It's going to make sense in a second and then I'm going to tape it before when I wasn't taping this and I actually just put the Teflon paper over it. It would just move. So what I do is just take a piece of heat tape. Now that I know I have it in the spot that I want it and I just do that just like that. So I'm going to take another one and then just show you and do the same thing. So I have this one. There's really no right or wrong. Just depends where you want to put your image on here. So it's like that. Same thing. I'm going to carefully take it. Place it down flatly. Let me adjust my camera for you guys really quick. I'm just going to put it down a flatly on here and then just take another piece of heat tape and go over. You can do like four or five or six of these at a time. Whatever will fit on there. So depending on which way you want your image to go. Kind of like it that way. But just like that. Make sure that you put your image up. So I'm going to put it down there and that you don't have to do this. This is just what I was doing and it just helps a little bit to keep it in place. But again, it's just a little trick if you wanted to do it. So I'm just going to do one more on here and show you really quick. Take another one that I don't have and then just do the same thing. Take the image and line it up. It just helps that you have a little bit of a border around it. I think if you did it the exact size, it would be a little bit hard to kind of figure out where you want to put it. But like I said, that's up to you how you want to do that. So I'm just putting a piece of heat tapes not necessary on there. Take my Teflon tape. I'm sorry, my Teflon paper. Put it over here. See, when I did this before and I put the thing on here very carefully even, it still moved all my stuff. So I didn't like that. That's why I have the tape on there. So watch this. So this I love this part. So I'm going to close this and it automatically goes down for me. And as you can see, it's counting down the time. So I have 42 seconds, 41 seconds, so on and so forth. When it's done, it's going to automatically open up top and I can just pull it out and it works so simple. There's no pressure that you have to figure out on here. You just need to know the time and the temperature, which I told you. So yours might be thicker or thinner than mine. So please test them out. Kind of see what works for you. But I have mine at 385 for the 45 seconds. So I'm going to let this count down and then I will show you the next step when it's done. All right. So now it's done. It's automatically going to lift up. Didn't do anything. Let some heat out. You saw that. So it's going to be very hot. And then I will show you what this looks like. You can put a pair of gloves on if you want, but this isn't too bad of a project for me to worry about burning my hands. So I'm just going to carefully take this tape off. That one was pretty. I'm going to take all this off and I will show you what they look like. Just be careful that you don't burn your fingers on this. Wow. That is a very vibrant color on there. Oh, look at that one. That's showing up how vibrant that is. My goodness. These beautiful watercolor ones. Wow. All right. So I have those now. I'm going to take the same exact image and I'm going to match them up but do it the same way. So I'm going to take the back side of this and to put it on top of here and it's going to kind of match up where I was with the last one. And then do the same thing with the tape. Again, you don't have to do this. It just kind of helps my stuff stay in place. And another thing I didn't mention is I have one of these Teflon papers on the bottom too because I'm trying to keep my bottom of this. So it doesn't that doesn't bleed at all. So I'm just going to take my next images, match those up and then I'll do the same thing. But you want to make sure that you throw these pieces away that you don't try to reuse them because all the ink is off of those. You want to toss those out. Let me just match these guys up really quick. So here's this one. You can see how the color difference is how this is very light and how vibrant this one came out. So I'm just going to take it, take the white part and put that on top and then just flip it. Very simple. I love this one. This one has to be my favorite one. So I'm going to take the same image and I'm going to put it on the back here. So I'm going to take the same image. Make sure I do it flat side, the white side down on it and the color side. Flip it upside down and then just put my tape on there. And believe it or not, I cannot find this other green one. I'm not sure where I put it. So I will have to wait to redo the green one, but we're going to show you this one really quick. So then you want to take your paper. You can use a butcher paper even if you wanted to, but you want to sandwich them together. All I'm going to do is close this automatically goes down for the 45 seconds. And then I'll show you the next part after this very simple. Very easy to do cost me probably 20 cents to create the whole thing. And I'm going to be selling these. I'm going to be trying them for $4 a piece. We will see because it's my time and my supply. So I'm going to try it at $4. I will let you guys know if you've done these before or if you know anything about these, leave it in the comment down below. How much you've seen them before, how much you would sell them for, what images you would like to see on these. Like I said, I did my dog one and that one turned out really, really amazing. So this works really well. So I'm going to let this finish up and then I'll show you the last step to this one. All right guys, so here comes the fun part. So I finished these. This is what they look like. They are double sided and actually funny story. I'm going to tell you the truth because I always tell you the truth. Remember when I said I couldn't find the green one? Well, I kind of messed up on that one. I did one side green and I thought I flipped it, but I didn't. So I have the one side green here and then I have this one here. So I kind of messed up on that one. But you know what? That's okay. Because when this spins in the wind, it's going to look kind of neat that it's double sided. But then I also have these that I cut out. They thought they were really cute. And then oh my goodness. Look at how vibrant these look. My daughter absolutely loves this one. She wants to keep it. And here is the one that I did of my dog Ray. I thought that turned out really, really cute. My daughter is going to love to put that in there.  So I'm going to show you that really quick. So here's how it comes. I will show you how it comes. It's not pretty cut. You have to cut this yourself. So you want to find. That's the fun part of it. You have to find one of the ends instead of just clipping it. I wish they would have actually given you pieces already cut because it's a trick to finding the beginning of one of these. Oh, here we go. So I found the beginning. What I actually did before is I just take it and I double it and you can make it how short, how long, whatever you want. But I just kind of guesstimate it and then I cut it and super simple. So you want to match up the ends here and then take your part at the top that looks like this and then you just stick it right through the hole and then you pull it through and then you have it on both sides. So it's like that and then you have this side. So you want to match these guys up as best as you can. Open this part up and put those two pieces through. So just like that. So it looks like that.  So you want to pull that kind of tight and then taking these two pieces make sure they're straight and they're both going the same way and then tie it as you would like a balloon or something and then just take it and pull your knot really, really tight.  Get as close as you can to the knot. So that's why I say get really, really tight with it. So it looks like this. Here's a double-sided one. But it looks just like that and then this is very, very stretchy. So you can take this and put this around your revier mirror. So just like that. So the best thing about these is you can spray it with whatever fragrance you want. Like if you have a perfume that you really like, a bath and body work fragrance that you really like, you can spray it. Or what I like to do is actually take fragrance oils and if they're clear, you can take them actually and drop them on there. Or I would suggest going to the dollar store, getting one of those little pump bottles that you spray, put some water in the spray bottle and then put a couple of drops of whatever scent you like and then you can always just go back and spray it with whatever you want. So let's say I did this with lavender and then the lavender stopped working after a couple weeks and I wanted to use, let's say grape. I would just go ahead and spray it with something else. As long as it is clear, it won't go on here. So you want to make sure that your fragrance, whatever you're using is clear. So these were really fun to make. I have to finish these, put the top on this, but they were super fun to make. I have these other ones already done with the ties on the top. But again, this has to be my favorite one. Just how vibrant the color looks on here. So guys, I hope you like this video tutorial. Give me a thumbs up, hit that subscribe button. If you're not subscribers of mine already, I'm going to be doing a ton more videos on sublimation and cricket and just every kind of craft you can think of. So look down below. I leave all the supplies down there. I leave any questions that you might have and I will see you in the next video guys. Take care.


Hey, what's up guys? My name is Brian. We're continuing our HTML tutorial. Now in this tutorial, I'm going to teach you basic text manipulation. This is one of the easier things, but it gives a lot of bang for your buck. The first one we'll show you is how to bold things. To do a bold, you just enter the bold tag, which is just a B. Whoops, got those mixed up. Let's save that. Flip back over to our web page. As you can see, I bolded the word hello. Now, one thing beginners often have a hard time with is, like me, they forget that slash. If you forget it, you're turning the tag on and not turning it off. What do you think's gonna happen? Well, if you guessed everything's gonna be bold, you'd be correct. You see how everything's now bold. Let's go ahead and fix that by turning the tag back off. Now, another one is italics. Instead of a B, it's an I. Just change that, save it, refresh. You can see how the word hello is italicized. You notice that world is not italicized. The reason for that is we've turned that tag off right here. Now, another one would be underlining. It's just a U. You can see the word hello is underlined. World is not. One thing you can do is you can mix and match these tags. For example, you can make something bold underline. Just remember to turn it on and turn it off. And let's go ahead and italicize the word world. Now, remember to save every time when you go to refresh this, otherwise you won't see your changes. Here you can see we have a bold underlined and italicized. Another thing you can do to manipulate tags is use what's called the font tag. The font tag is very simple, but very powerful. Just remember to turn it on, turn it off.  As you can see, nothing's happened. You have to give what are called attributes. So we'll say we want the color to be red. Now you can choose any color you want. There are standard colors. Typically, they're friendly names like red, green, blue, black, but you can also use a hex code. For this, we're just going to stick with the name. We don't want to confuse anybody. Now save your work, flip back to your web page, refresh it. You can see we now have hello world and red. We're not done though. The font tag actually has extra attributes. You can define, for example, the color, the size, one being incredibly small as you'll see here. And it goes up to, I think seven is the highest, but we'll find out. Go ahead and refresh this. So you can change the size of these things, which is actually pretty neat. Now let's see what happens when we just split this up a little bit. Notice how we have a couple lines here. Hello world. Take a minute and think about what this is going to look like. Do you think hello and world will be on two different lines? Let's find out. No, the reason is we have not told it to put a hard return in here, but you notice how hello follows all of the text processing we put on it. It's red. It's size is seven. It's bold. It's underlined. World is just italicized. Now I'm going to show you how to force this to go onto separate lines. You enter what's called a hard return or a break. The break command does not have an off tag. It's a single tag, no off. For clarity, I'll just remove that extra one and you can see what's going to happen here. Every time you do a break command, it shoves it down one extra line. There you go. We have hello with all our fault processing, a hard return, which is not visible, and then the word world italicized. Now you can enter as many of these break commands as you want. It's actually very common to enter a couple of them right in a row just to move the line down further. Another thing you can do is change the font or the face. We'll just use aerial. That'll tell it to use a specific font face for that text. Now remember, just this is getting manipulated. The word world is not. For clarity's sake, let's actually just copy this tag and be sure to turn this off here. Now we've got two separate font tags. Just space this out so you can see what's going on. Let's make the color blue and let's give it a size of three. Now we have two font tags. We're turning the font on and off, and then we're turning the font on and off again. One word will be red, one word will be blue. And there you go. Thank you for watching. I hope you find this entertaining and educational. I'll try to make more videos as time allows.


Welcome to our lecture on probability distributions. Let's talk about probability distributions. A probability distribution describes the likelihood of various outcomes for a, let's call it an experiment, really a stochastic process. For example, suppose you want to know what is the likelihood of getting five heads and five coin tosses. Okay, five heads is one of the outcomes. Of course, you can get zero heads, you can get one head. So the way you would determine that is you'd be using what's called a discrete probability distribution. The one we'll use is the binomial. You can actually get the probability of five heads, four heads, three heads. Notice these are discrete. You can't get 1.67 heads. Now sometimes we're working with continuous measurements. For example, you want to determine the probability that an American adult weighs between 200 and let's say 220 pounds. Or you want to know what is the probability you have to wait between 20 and 40 minutes for a bus. Now wait and time, as we know, those are continuous measurements that we need to work with a continuous probability distributions. The thing, the one way of knowing something is a probability distribution. If you look at all the outcomes, add them up, and it should add up to one. Because the sum of all the probabilities of every outcome is one. Let's start out with a simple example, the toss of a single die.  The number is one, two, three, four, five, six.  So that's our random variable with random outcomes. And assuming the die is fair, not biased in any way, the outcomes are all equally likely. You're just as likely to get a one or two or three or four or five or six. And so each one has the probability of one out of six, probability of one sixth. And if you like, you can call it a uniform distribution. But right now, all we want to see is that we can create a distribution by listing all the possible outcomes of a particular stochastic process. And alongside of each outcome, the probability of getting that outcome. There are two types of probability distributions, depending on the type of the random variable that you're studying. The random variable, remember, is the variable whose value is determined by chance. It's the outcome of a stochastic process, of a random process. The random variable, as we saw in the previous slide, can be either a discrete random variable or it could be a continuous random variable. For a discrete random variable, we would have to use a discrete probability distribution. And for a continuous random variable, we would have to use a continuous probability distribution. The reason is we can't use one for the other. A discrete random variable can only take on very specified, distinct values. And a continuous random variable can take on continuous values, can take on any value within a continuous interval. So naturally the way we describe the probability distribution is going to be different for these two different types of random variables. There are three major discrete probability distributions. One we're going to learn in this course is the binomial. And from the word buy, you know there are only two outcomes, like defect and not defect. So sometimes companies will actually study how many defective parts are there in an electric car, for example. Now again, the binomial distribution can answer the question. Now another discrete probability distribution is the hyper geometric distribution. Now this one is especially important if you're going to be a card shark and you like to play poker. The two outcomes without replacement and you could for example determine what is the likelihood of getting four aces when you play poker. In any case, we won't be doing it in this course. The third one is the Poisson distribution, which probably won't be covered in this course, but you may see it in other courses. So what is the Poisson distribution? And you have a discrete random variable and a continuous interval. For example, the number of defects per square yard of material. We can also simply enumerate the possible outcomes and indicate the probability for each one. We're going to see an example in the next slide. The probabilities for all the listed outcomes must add up to one. That's what makes it a probability distribution. You could have a mathematical formula to represent the discrete probability distribution and to compute probabilities. Those are the ones that we talked about before. The very well known discrete probability distributions. But you could also construct a discrete probability distribution by listing all the possible discrete outcomes from this random process. And for each outcome, you would have a probability associated with it. And you see a little example on the side of the toss of a die, which you saw before, but now it's looking like a probability distribution. So if we look at the discrete probability distribution more generally, we can see that there are three major properties that apply no matter what you're looking at. All the probabilities taken together sums to one. If you're enumerating or if we could do it differently with a formula, but the sum of all possible probabilities of all possible outcomes has to be one. That's your universe. That's 100%. If you come out to something less than one, it means you forgot an outcome. If you come out to something more than one, that may be a bigger problem because it probably means that your outcomes are not mutually exclusive. Imagine in the example you see here that I have not only the outcomes listed, but the outcome, the die comes up odd, which would include one and three and five. So there's overlap there. That wouldn't be overlap with the individual outcomes that are listed. So the sum has to be one, not less than one, not more than one, but it has to be one. The probability of any particular outcome is greater than or equal to zero, and we know from one, from number one it has to be less than one, but we also know that from the rules of probability that we learned in the last lecture. So any probability has to be between zero and one. Not less than zero and not greater than one. And finally, when you substitute a value of the random variable into the function, assuming you're using a function, like the well-known distributions on the previous slide, then your function computes to the probability of getting that value. Another thing you can see if we had to give you another type of discrete probability distribution, other than what was listed on the previous slide, the probability distribution for the toss of a die is a uniform distribution. It's a uniform discrete probability distribution because all the outcomes are equally likely, and since there are six of them, every outcome has the same probability as everything else, one, six. Let's talk about the expected value. What is an expected? It's very much like a mu. It's a single average value that summarizes the probability distribution. The best example, and the easiest one perhaps, is what do you expect to get when you toss a die? Only six outcomes. You either get a one, a two, a three, a four, five, or a six. They all have the same probability, one, six, one, six, one, six, one, six, one, six.  Let's see it a different way. Let's say we have a class of 40 students. Each one tosses a die, and then we average out what they got. The expected value is 3.5. How do I know that? Well, just common sense. You'll probably get as many sixes as ones, and the average of six and one of course is three and a half. You'll get as many twos as fives. The average of two and five is three and a half. You'll get as many threes and fours, as many threes as fours, and the average of three and four is three and a half. On average, you'll get three and a half. Now of course, if you toss a die, you're not getting a three and a half, but this is what it averages out if you keep doing this over and over again. That's called the expected value. We don't really want to use intuition in order to figure out the expected value of a distribution. It's easy with a die, but with other distributions, it may not be that easy. We want to be able to use a formula that's called mathematical expectation in order to compute that one average value, that long-term average, over the whole distribution. Let's call our random variable x. x takes on values, and we know that when we construct the discrete probability distribution, every value is associated with a probability. If we take each value, multiply it by the probability, and then sum all those products up, we end up with the expected value for the distribution. You can see the formula there. The expected value, e of x, is the sum of x times p of x over all values of i, over all outcomes of this random variable. This is going to end up being a weighted average. When we saw the distribution of the die, it's actually a straight average, but that's because all the outcomes were equally likely. When they're not all equally likely, you end up with a weighted average, and that's your expectation. That's the mathematical expectation. That can be used as an average, and it actually computes to the parameter of the distribution, the parameter called mu, the mean.  Remember the formula is, we're using d now, the sum of the d, that's the outcome, times its probability. You take one times a six to get a six, two times a six is two six, three times a six is three six, four times a six is four six, five times a six is five six, and six times a six is six six. Now you have to sum that column. That's the last column there, and you get 21.6. The 21 divided by six, 21 divided by six is three and a half. This proves that the expected value when you toss a die is 21 over six, or 3.5. We've looked at the expected value of the mu in effect. Just like there's a mu, this should be a variance in the standard deviation. We have sigma squared, the population variance, and we have a formula, and you can just follow the steps and you can see that we ended up with a variance when you toss a die of 2.92, and the standard deviation is 1.71. The expected value is an important one concept, and now we're going to learn about expected monetary value mostly used in decision making and business. You want to know what is the value, a lottery, or in fact when you invest in a product, you know there's a certain probability you'll make so much money, or a different probability for making a different amount of money. We can always compute the expected value using the formula which we just learned, the sum of the x times the probability of x. So here's a game. Let's call it a lottery. What are you going to call it? It could be a business investment, and the three outcomes $300, $120, and zero. And suppose your subjective probability, you believe that a one-third chance, you'll make a $300. It doesn't have to be subjective, it could be an actual roulette wheel and it works out that there's a one-third chance of $300, a one-third chance of $120, or a one-third chance of $0. On average what do you think you'll be making each time? Remember, one-third of the time you'll make $300, one-third of the time you'll make $120, and one-third of the time make $0. So using the formula $300 times a third plus $120 times a third plus $0 times a third, you end up with an expected monetary value or expected value of $140. That means on average you're going to make $140 per game if you play this game for a long, long time. This is very important in business decision making, or if you're looking at the lotteries and the value of that lottery. We have another example of EMV. In this example, we're basing our decision on the toss of a coin. Of course, I don't think that's the way you're going to do it when you are the decision maker in business. But this is framed as a game, so you can think of it as a game, or you can think of it as a business decision. The coin is tossed. If the coin comes up heads, you win $100. If the coin comes up tails, you lose $50. What's the expected value of the game? Okay, why do we need to know what the expected value is? Well, maybe you have to pay to play. The answer is the expected value of the game is $25. How do you get expected value? The same way we got it before. The outcome times the probability, $100 times one-half is $50. Negative $50 times one-half is negative $25. You add those up and you get $25. Why $25? What does that mean? Remember, mathematical expectation which holds for expected monetary value is the expected long-term value of this game or this decision or this lottery. What that means is that the assumption here is that you're doing this game or making this same type of decision repeatedly under identical circumstances many, many, many, many times. Half of the time you'll win $100 approximately. Half of the time, approximately, you'll lose $50 on the toss of the coin. So that over the long term, if we take the winnings and we take the losings and we add them up and divide by the number of tosses, over the long term you can expect to make approximately $25 per toss. So if you had 100 tosses, 50 of them you'd say you won $100, 50 of them you lost $50. So over the 100 tosses you make $2500, divide by the number of tosses you have $25 per toss, per game played. So no matter what anyone tells you for this particular game, don't pay more than $25 per toss to play. Example three is very similar. Instead of tossing a coin we have probabilities for each outcome, for each of three outcomes. One, it's framed as a type of a game. One outcome is that you might lose $100 and there's a 1 quarter chance of that happening. You might be ahead, you might win $80, there's a 1 quarter chance of that happening. The rest of it, half of the time, you come out even. You don't lose, you don't gain. So the question is as always how much should you be willing to pay to play? And for that you have to figure out the expected monetary value for playing one game. And you see here it comes out to negative $5. How did that happen? Negative 100 times 1 quarter because that happens 1 quarter of the time. Plus 80 times 1 quarter, zero times anything is zero and when you add that up you end up with a loss of $5. How do you pay negative $5 for playing a game? I think I'll leave that up to you to discuss. This example might remind you of the lottery ticket and in fact that's what it represents. We'll look at a lottery ticket and you get one chance in $5 million of making a million dollars, the jackpot, four chances in $5 million of making $100,000 and the rest which would be $4,999,995 over $5 million of making zero. Remember all the problems have to add up to one. And somebody says well how much is this lottery ticket worth? Well if you do the formula, a million dollars times one over $5 million which is $0.20, $100,000 times four and $5 million that's $0.08, zero times anything is zero, add it up you get $0.28. Basically this lottery ticket should not be worth more than $0.28 now of course if they charge $0.28 to the lottery ticket they break even. So assuming you're buying this from the city or state they're going to charge you like $1 for this lottery ticket for the privilege and if you don't know statistics and you don't realize the expected value is a lot less than $1 you might be foolish enough to pay $1 but clearly according to mathematicians the expected value is only $0.28. Here we have another lottery, it's not that different from what we had before although the values work out a little differently, not much but a little. There's a $5 million payout the probability of that $5 million payout is how many zeros are there? $0.0001 it's probably very close to a typical lottery ticket which may be even worse. The usual lottery tickets are probably much worse than this. If you look at EMV, expected monetary value, the expected value of this lottery is $5. You shouldn't pay more than $5 for a ticket. Tickets will always be more even more than $9 because people hold out hope that they are going to be the one who wins. Another reason we don't use expected monetary value typically as an individual when we're buying a lottery ticket is that most of us don't do the same decision under identical circumstances many many many many times because that's what EMV represents, expected value represents that type of decision. When people pay $9 or $10 or $20 for a lottery ticket like this, they're not using EMV for their decision making. Same thing with slap machines in a casino. Any kind of gambling is the game is going to look something like this. Slap machines are much worse. They're loaded much worse than this. You know who uses EMV in gambling? The house. The lottery commission that sets up this lottery. The casino administration that sets up the probabilities for the payouts and the slap machines, they're using EMV. They know exactly what they're making over the long term for all of their decisions. What do people use if they're not using EMV and making decisions like this? You may learn this in another course. You're not going to learn it in this course, but just so you know it's something called utility theory. A continuous random variable, if you recall, is a random variable that can take on any value inside of an interval. If you have a variable like, let's say, weight of adult males, it's not going to go down to the negative numbers. It's not even going to go down to zero, but between some reasonable values on the low side and on the high side, like maybe 75 pounds to 700 pounds, right? The random variable can take on any value. It doesn't have to be a whole number. It's not specific, distinct values. You can't enumerate. You can't create a continuous probability distribution the same way you can a discrete probability distribution. You can't just list possible outcomes and say, well, you know, what probability is associated with this? We have to ask for probabilities inside of an interval. I can't ask a question like, what's the probability that an individual adult males weight will be exactly 95 pounds? I can't ask that question because what does 95 pounds mean? Is it 94.99999? Well, that's not 95. It's a continuous random variable. The probability that a continuous random variable takes on a particular value over all the infinite possible values it could take on, it's zero. The probability that a random variable, weight of an adult male, will be exactly 95 would be that one possible value divided by an infinite number of basically it's zero. So we can't do it that way anymore. We can't ask what's the probability that our random variable will take on any one particular value. We have to ask our questions our probability questions about intervals. What's the probability that our random variable will take on a value, just looking at the picture here, a value between zero and one. There we've got looking at the picture we've got a continuous random variable with zero at the middle, zero is clearly the mean. And we want to know what's the probability that this random variable takes on any outcome between zero and one. If we consider the entire area under the curve, that curve is the function of that random variable, of the probability of that random variable. If we consider the entire area under the curve as the sum total of all of the probabilities well that's one. Remember the sum of all the possible probabilities is going to be one. Another way to look at it is if we take the definite integral, the shaded area under the curve, sum it up and divide by the total area under the curve for the entire interval of the entire random variable from let's say in this case negative infinity to plus infinity, we get a proportion. We get a proportion of time that this random variable will take on values inside that shaded interval. Well that's great because if we could define a proportion we can consider it a probability. A proportion is the same as a probability is the same as a percentage in the case of looking at continuous probability distributions. From the previous slide remember that for any continuous probability distribution, and there are a lot of them, some we will study and some we will not study, the sum total of the area under the curve is always one. That's the universe. The universe is 100% of what you're looking at. What are we going to look at? Well definitely we're going to do a lot with the normal distribution. That's a continuous probability distribution and it's going to take up quite a bit of time in our course. A variation of the normal distribution, the standard normal as another one we're going to do a lot with. Another variation of the normal distribution, the student's t-distribution the chi-square distribution and the f-distribution are optional statistics. So depending on the what course you're taking you may or may not be doing it in this introductory statistics course this semester. And another one that will not be done in our course and it's a good distribution that's used a lot but it's not used in an introductory statistics course is the exponential distribution. The exponential distribution is also a continuous probability distribution and just to give you an example if you've ever seen a learning curve or a decay curve those are examples of exponential distribution. Thank you for attending our lecture. In this lecture we introduced the concept of a probability distribution for a random variable. If the random variable is a discrete random variable we need to look at a discrete probability distribution and we're going to go further in depth in one example of a discrete probability distribution, the binomial distribution. If the random variable is a continuous random variable we need a continuous probability distribution and we're going to go further in depth into mainly the normal distribution in that lecture and in further lectures where we look at variations of the normal distribution. Practice, practice, practice, do your homework, do every problem you can find. There are lots and lots of problems in various places on the website. Even when you did this lecture I hope you paused it at the appropriate time and did the problem first and then listened to the narration. The only way that you will do well is to do very, very well in this course and in fact not only do well but you'll find it easy is if you do lots and lots of problems. Practice, practice, practice.


This 2020 Ram 1500 crew cab six foot four inch long box is stock number 20 T156A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used Ram and light-duty truck headquarters today. We are checking out this 2020 Ram 1500 and this one has the 5.7 liter V8 Hemi motor with the e-torque assist. It's going to give you 22 miles per gallon on the highway 17 city and an average of 19 miles per gallon. This truck has been fully safety to and inspected by our service shop has a fresh oil and filter change. All the fluids have been checked and topped off and this truck is 100% ready to go. Billet Silver is the color we shoot all of our videos in 1080p 60 frames per second. So if you have HD capabilities on your computer tablet smartphone or television turn them on right now because it is definitely your best way to check out the look styling and options of the truck before seeing it in person. And if you'd like to subscribe to our YouTube channel in the lower right hand part of the screen is a gold subscribe button and click that and click all the bell notifications to see videos of our inventory that we do daily. Give a shout out to a couple subscribers up in Marathon County. Jake, Jared, and Mike. They watch regularly. Thanks for watching. Comes with the 22 inch painted and polished aluminum rims and it has Goodyear Eagle Touring 285 45R20 tires. And these tires have right around half the tread left on them. Front fenders in excellent shape. No dents or dings on there. Get the LED headlamps, the LED running lights, and the LED fog lamps. Front bumper is in excellent shape. I didn't see any dents or dings on there. And the hood is in great condition as well. No dents or dings on there. Get the chrome trim grille. And the passenger side front fender is in excellent condition as well. Passenger side rim. No major scuffs or scrapes on there. And as you go down this side of the truck, take note of how clean the body is, how reflective and mirror-like that paint is. We take these HD videos so if you are far away or even if you're close by, you just cannot make the trip down. But you're so interested in purchasing the truck, you can see the truck, hear the truck, and have confidence in the vehicle that you're looking at before you even get here. So when you do get here, there's absolutely no surprises and you can make a smart mind decision from the comfort of your own home. Down this side is very nice and clean. Didn't see any dents or dings. It does have the power boards and these back rims. This back rim is in excellent shape. And the back tires have just about as much tread as the front tires. Maybe just a little bit less. It does have the plastic wheel well linings, which keeps your frame and underbody looking really good. And once again, down this side, very nice and clean. Coming around to the back of the vehicle, rear bumper is in great shape. It does have the full towing package, which includes a receiver hitch, four pin, and seven pin wiring. And the tailgate is in really nice shape as well. I didn't see any dents or dings on there. I believe this is a Mopar factory tonneau cover. Spray and bed lining. You get the bed rail system with the cleats, the LED bed lighting. The tailgate has the spray and liner on it as well. And that has these shock down assist where it's never going to slam on you. LED tail lamps as well. And as you go down the driver side, just as clean as that passenger side, no dents or dings on the box. And for full disclosure, this back rim is in excellent shape as well. It does have the capitalist fuel fill, which is a really nice feature. Never get gas on your hands again. And this one does have the four corner air suspension where it'll actually level out the suspension for you. You can raise it. You can lower it for better aerodynamics or off-road capabilities. Heated mirrors, blind spot monitoring, directional signals, and blind spot mirrors. Get the entering ghost system. Check out those power running boards. They work really nice.  Now this is all leather as compared to the Laramie's where it has the suede inserts. So this has the full leather seats with the intricate stitching on it. You get the gray piping. Both these seats are heated, cooled, and power. Driver side lumbar factory all-weather floor mats. Those carpet pieces come out to make them true all-weathers. Auto headlamps, power pedals, telescopic steering wheel.  Get that leather bolster and the wood grain trim on the doors as well. These mirrors do power fold in. I always like showing both sides so that you know that both sides are working properly. We'll hop inside here and check out the instrument cluster and the radio. So you can see all the options on this truck. 21,526 miles. You do get a digital speedometer. Compass display range to empty and you can change those corners to read whatever you want. This is a special design on the speedometer for the limited package. I think it looks really good with the blue accents there and the 7-inch LCD display. You get the heated and leather wrapped and wood grain trim steering wheel. This one has cruise controls as well as adaptive cruise controls. Gear selector, Bluetooth and information center controls, push button start. 8-speed automatic transmission, push button four-wheel drive and your start-stop capabilities, which you get with that e-torque system. This one has the factory navigation system on the 12-inch Uconnect 4C radio. You have lane departure, perpendicular and parallel park assist, Harman Kardon sound system up there. Get the wood grain trim, the bronze trim on there and then this like finished metal look as well. Climate controls on the radio, climate controls in the radio. You can do your heated and cooled seat buttons, your heated steering wheel. You can also check out your backup camera. There's your backup camera and they made these HD so like they're crystal clear. Those grid lines are dynamic and you can zoom in right over your receiver hitch to get hooked up to your boat camper trailer the first time every time. And then you get all your different apps on here as well including projection manager where you can project your cell phone to the screen via Android Auto or Apple CarPlay. We actually did a demo on Android Auto if you want to check that out to see what you can all do with that system. And then it does have the surround view camera there as well and you can select your different cameras so you can do a wide view of the back, narrow view of the back, wide view of the front or a narrow view of the front just for better maneuverability while you're parking stuff. Down here you get your factory brake controller. This is your four-corner air suspension control so you can raise the truck up to off-road mode 1, off-road mode 2, normal height and then exit height on there as well. So there's four different settings, stability control, tow haul, your front and rear parking sensors. You do get two USBs, two USB-C's and an aux jack, your RAM charger for your wireless cell phone charge pad, 115 volt 400 watt plug-in.  This one's kind of outlined and trimmed in gold and says RAM limited on the back. You get the limited badge on the center console with that intricate stitching on there too. Passengers side dash, floor mat and seat are in excellent shape. I didn't see any rips or tears on that seat and this truck's never been smoked in. It has the power sunroof with the vista roof so the full second row glass roof and then the power sunroof in front. Home link buttons are up here and then you do get your map lights and controls for the sunroof, tailgate, SOS assist. All that good stuff has the auto dimming mirror and we'll check out the back seats and then start her up. Back seats are just as clean as front seats, no rips, no tears back here. It does have the latch child safety system for any child car seats you may have. Power sign rear window with the built-in rear defrost. You do get that intricate stitching and gray piping all the way around on the back here. These seats do fold up for extra storage. There is a collapsible load floor right there or a collapsible storage bin right there. You do get in-floor storage on each side with two removable containers. And these back seats are heated. These back seats are cooled. You do get a USB, two USBs and two USB C's and then another 115 volt 400 watt plug-in. You get that all-weather floor map back here. Back seats have the same trim and stitching as well. And you get the same trim and stitching on the back doors as well as the child safety locks. You will turn the truck on. Go check out all those LED lights. Of course, I'd like to thank you for checking out the video as well as all the people that subscribe to our channel. Under the hood, we have the 5.7 liter V8 Hemi motor with the e-torque assist. So when it has this big silver thing on there, that's when you know it has the e-torque. That's actually the electric motor right there. Once again, 22 miles per gallon on the highway, 17 city combined average of 19. And once again, this truck has been fully-safetyed and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. And this truck is 100% ready to go. There's your missing sticker. There's those LED headlamps and running lights. And to see more pictures of this truck or one of our other 450 new and used cars, trucks, SCVs, minivans, Wranglers, three quarter tons, one tons, you name it, we got to go to that website right there, www.summitauto.com. Full pictures and descriptions of every single vehicle from two locations all at www.summitauto.com. And if you'd like to check out more HD videos, you can go to www.youtube.com.com. Remember to like, subscribe, and share on this video and all the videos that you see there. In fact, in a second, you will see a link to subscribe to our YouTube channel in the upper left. A link to more Ram 1500 truck videos like this one in the upper right. A link to this vehicle on our website in the lower left and a link to one of our latest YouTube videos in the lower right. Click those check us out and we're super excited to help you with this Ultra Clean 2020 Ram 1500 Crew Cab Long Box Laramie Limited Package and Billet Silver Metallic. Thanks again so much for checking out the video and remember to like, subscribe, and share.