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Problem 21 Hard Difficulty

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 3

Answer

$F(x)=\frac{1}{63}\left(x^{3}-1\right), \quad 1 \leq x \leq 4$

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Video Transcript

for this problem were given a pdf probability density function, which is F of X and are asked to calculate the CDF, the cumulative distribution function. Uh, that is associated with this. Pdf to do that, we're going to start out We're going to start out by integrating are the left of axe with no limits on our integral with respect acts. So this is going to give us X cubed over 63 plus c right, Because we have no limits we're gonna get plus a constant. Our next step is to figure out what this constant is exactly. So there's a couple different ways Ah, that we confined this. I prefer to plug in the value on the right hand side of our interval into our equation. So doing that. See, that effort for is equal to four to the power of three, which is 64 over 63. Still, plus that constant. And then I'm gonna note that since we're on the very right hand side of our CDF, what are cumulative distribution function is supposed to define is how much probability we've encountered as we move left to right along our number line. So when we're on the far right hand side. We should have accumulated all of the probability up to that point. What should just be want, right? And so if we go ahead and solve this equation, will see that see is going to be equal to negative one over 63 because that's going to give this function value of one. So how we typically show the total CF Ah, is we're gonna take this equation with R C plug then and put it here. Ex Whoops. That should be an X cubed over 63 minus 1/63. Uh, and this is for all X in our interval. This is what we want our function to be. So, any X between one and four. Any excess that come before one. We actually just want the value to be zero. Because if we haven't gotten into this interval yet ah, we haven't found any probability. And likewise, any X is greater than four. We want to be one because we've already accumulated all the probability. So this function cap affects our CDF in its defined on the entire interval, even though our probability is only to find from 1 to 4. So this whole thing is what the final answer should be

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