 So in this final video from lecture 24 about probability and integration, I wanna introduce the notion of mean, or what's often called the expected value. So if we take a probability density function, y equals f of x, and we are just treated like any other function, we might ask what is the center of mass of this thing? Well, because every probability density function, the area under the curve is always equal to one, finding y bar is not so exciting, right? But the distribution is dramatically affected by x, right? So x bar is significant when one, when one is considering a probability density model, right? And so if we calculate x bar in this situation, you're gonna get one over a, the integral of x f of x dx. And notice that because it's a density function, the area under the curve is always gonna equal one. So this x bar always simplifies just to be the integral of x f of x dx. And so this right here represents the average or the mean, or like I often, like I said before, this is called improbability expected value. You get e of x where x is the random variable, sometimes they call it mu, mu is for mean, it's the Greek letter kind of like for m, it's not a Pokemon in this context. And so the expected value is gonna be the integral from a to b of x f of x dx. And this is what we call the expected values because this is the middle of the data. This is the one that's in the center of all of the distribution. And so let me show you some examples to give you some context of what it looks like. We saw this example earlier in our series, f of x equals three over 26 x squared. It's a density function whose domain is one to three. The expected value here is just a, we can find the calculation here. And so what this would be is one over the area, which again, that's always one here since it's a density model. You're gonna integrate from one to three x times three over 26 x squared dx. Take out the three over 26, we have to integrate from one to three, the x cubed dx. That does change the calculation from what we did before. We're now gonna get x to the fourth over four as we go from one to three like so. So we're gonna get three over 104 times that by, we're gonna get three cubed, or sorry, three to the fourth, which is an 81. Whoops, sorry about that, 81. And then minus one to the fourth, which is one. That's gonna be 80, so we get three times 80 over 104. There are some common fractions, some common divisors in the fraction can simplify. You get 30, 30 over 13. But again, decimals are really appropriate here. We're gonna estimate this to be 2.308. 2.308. What does this number mean? How should one interpret this number? What this tells us is that if you are looking, if you just pick a random place on Earth, you should expect to go 2.3 kilometers before you find a bird's nest. You could find it closer, it might take a lot longer to find it, but what you can expect is you should expect to search two and a half kilometers before you find that nearest bird's nest. As another example, find the expected value of the random variable X whose probability density function is given as f of X equals three over X to the fourth as X is greater than one. One can show that this is a probability density function, I'm not gonna do that here. I just wanna compute the expected value. The expected value is gonna be the integral from one to infinity of X times three over X to the fourth DX. There's a little bit of cancellation here. X cancels with one of these, so we get a three. And so then computing the antiderivative, we're gonna get three over X squared because we raised the power and then we divide by the power which is negative two as you go from one to infinity right here. And so because there's a negative value, I'm actually gonna switch it around. So we're gonna take three over two X squared as you go from infinity to one. And so this is gonna be three halves minus three over infinity, that's of course a zero. So we end up with three halves are 1.5. So this one, I haven't given you any sort of physical interpretation of what this model is measuring, but what we can see here is that X will randomly be assigned any number from one to infinity. And if I had to guess, I would expect it to be 1.5. That's the middle of this distribution here. All right, let's look at one more example that we haven't seen before. A recent study has shown that airline passengers arrive at the gate with the amount of time measured in hours before the scheduled flight time given by the probability density function F of X equals three quarters two X minus X squared from zero to two, that is people will show up anywhere from zero to two hours before the flight turns up. What is the expected value here? One can also show that this is a density model and a probability density function. I'm not gonna do that here. If you take E of X, we're gonna integrate from zero to two, three fourths, X times two X minus X squared DX. Distribute the X, take out the three fourths. We go from three fourths. We're gonna go from zero to two, two X squared minus X cubed DX. Finding the anti-derivative here, three fourths, we're going to get two thirds X cubed minus X to the fourth over four as we go from zero to two. Plugging in zero makes everything disappear. Plugging in two is a little bit more interesting. Three fourths right there. When you plug in two, you're gonna get two times two cubes. That's a 16 over three. You're gonna subtract from that two to the fourth, which is also 16 over four in this case. I'm gonna factor out the 16. So we get three times 16 over four. Four of course does go into 16 four times. And then that leaves behind one third minus a fourth, which case I'm gonna write this as four twelfths. And then we're gonna take away three twelfths like so. And so that ends up with three times four times one 12th, like so. And so this will simplify just to be one. And there are units attached to this thing. This is one hour. So what this tells us is that this result indicates that passengers arrive at the gate on average an hour before the scheduled time of departure. That's how we should interpret this expected value. It tells us what we can expect. That's what average is all about. What should we expect? It doesn't tell us what will happen. There's no guarantee here, but it does give us an expectation. And so that'll bring us to the close of this. Just short trailer of how calculus and probability are really connected to each other. If you want to learn more about this, I would recommend you take a class like math 3700 at Southern Utah University, a probability statistics class that's based upon calculus. This is a standard class at all universities. I recommend you take some, you learn some more about that. You can see how this calculus you've been learning in this series is very appropriate to many other real life problems. We focus a lot on physics in this class, in this series, but we can also see some other applications that we saw with these probability models here.