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Numerade Educator

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Problem 24 Hard Difficulty

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 12

Answer

$F(x)=\frac{1}{211}\left(x^{5 / 2}-32\right), \quad 4 \leq x \leq 9$

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Video Transcript

okay for this problem were given the PDF or probability density function for some distribution and were asked to find the CDF. Ah, typical notation for the pdf is little FX. Typical notation for CDF Cap FX. Our first step in calculating capital effects is going to be just integrate Ah little F of X with no limits and are integral with respect acts. Yeah, and so, getting this, we're going to get X to the five over to here. Ah, and then we'll get one over to 11 here, and we don't want to forget. Since we don't have bounds in an integral we get a plus e. It's very important for this question. Our next step, then, is going to be to figure out what that seat is. And so there's a couple of ways to calculate it. I prefer to plug in our right endpoint eso. In this case, that's nine into our function f of capital life of nine. It's a plugging a nine in for this ex. I'm gonna start by taking the square root of it to get three and then taking three to the fifth power on. So it's gonna be nine tops nine times three. It's gonna be 81 times three, uh, which should be to 43. Okay, so we're gonna get to 43 over 2 11 plus c. And since this is the right hand side of our interval, we know that we've already moving from left to right. Crossed all of the probability that's in this, uh, region. And we want our probability always to add up to a total of 100% or one. So we're gonna set this equal to one and solve for the sea. Ah, here. We can easily see that r c should be negative. 32 over 2 11 Right. That'll give us a value of one here. And so our final answer we're going to go ahead and write out capital fax like this in the middle here. We're gonna take this function we just calculated and plug in our value of see that we also just calculated. And this function is our CDF for X only when it's on this interval, when X is between four and nine. We also want to think about one X is less than four. And when X is greater than nine, right when X is less than four. We have not crossed any of the probability yet. None of the probability happens before four. So we've accumulated zero probability when X is greater than nine going from left to right. We've already walked along all of the probability. So we've already accumulated 100% of it. So after dying, our functions just gonna be equal toe one on DSO. Your final answer would be this whole thing Ah showing that access to find not just from 49 on the entire number line, even though our probabilities only between four and nine.

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