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Problem 22 Hard Difficulty

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 4

Answer

$F(x)=\frac{1}{98}\left(x^{3}-27\right), \quad 3 \leq x \leq 5$

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Video Transcript

Okay, So for this problem, we want to go ahead and take our pdf that were given our probability, density, function little f of x and calculate cab F of X, which we used to denote the cumulative distribution function. And so how we're gonna do this is we're gonna start by just integrating our little F of X with no limits on our integral. So that should give us one over 98 x cubed plus a constant. Because we have no limits under control, We're gonna get that plus C on the office. And so from here, our next step is to figure out what this plus C is going to be. That makes us a valid CBF. There's a couple of ways we can find this the way that I prefer is to plug in the right end point into our function. So for here, we're going to get 1 25 over 98 plus c. And then I want to note that at the right end point of valid CDF should give us a value of one right. Another way to think about this would be playing in the left endpoint and noting that should give us a value of zero. So the value of see here that's gonna make this equation positive one is gonna be the, uh, correct value. See, for our CDF and just looking at this, we can see this should be negative. 27 over 98 right? Is that how we want to actually right? Our final answer. Look, something like this, we have f of X in the middle spot. We're gonna put the function that we just calculated 1/98 x cubed, plus our constant, but we're actually gonna fill it in. So minus 27/98. And this is defined for all X on the interval. So this is defined for three less than or equal to x less than equal to five. Right? Any accident interval. This is the function we want. Then we're also going to define it for any X less than three and any X greater than five. So since this function talks about how much cumulative probability we've gathered as we move left to right on a number line any time that were less than three, we have yet to encounter any probability because it's defined toe only be between three and five. It's all we want here is zero and likewise after five, we've already accumulated all the probability. There's no more past five. So we should have lips so sloppy we should have one for all ex care than thought. So this is the final answer for our CDF, and it is to find on the whole number line, even though we only have probability between three and five.

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