 In this video, we provide the solution to question number three for practice exam number three for math 1220, in which case we have to compute the probability as our random variable X varies between zero and two, given the probability density function that F of X equals one half times one plus X to the negative three house power, and for which this density function is defined on the interval zero to infinity. Now, notice if you take the integral of F from zero to infinity, you're gonna get one, but to find the probability, we just have to compute the integral from zero to two of said function, one half times one plus X to the negative three house power, for which you can pull that coefficient of one half out in front, then you have this one plus X to the negative three over two power there. You could do a U substitution X, U equals one plus X, DU then equals this DX, but as you don't have to change the differential at all, I'm just gonna recognize that my anti-derivative can be computed as, again, the one half comes out front. You're gonna raise the power of one plus X here to be, you add one to negative three house, you get negative one half. You're then gonna divide by negative one half, which actually makes these one halves cancel, you do have a negative sign still, as you go from zero to two. Now, because of the negative sign, I'm just gonna switch the boundaries around so that we get one plus X to the negative one half power as you go from zero, excuse me, from two to zero now. Nope, I didn't switch them, zero to two. One more try was necessary there. When you plug in zero, you're gonna get one plus zero, which is one to any power is gonna be one, so the first one's just gonna be a one there. And then you subtract from that, if you plug in two, you're gonna get one plus two, which is three, three to the negative one half power. I might think of that as one over the square of three, but that would be the exact probability. You're gonna put that into a calculator, one minus one over the square of three, you should then end up with point four, two, two, six, et cetera, that's actually sufficient for us. That gives us F is the correct answer, it's about 42%.