Finding the PDF of Y = X2 with Given E(X) and E(X2)

[Quincy]

Homework Statement

random variable of X has pdf: f(x) = (3/16)*x² from interval (-2,2). Also, E(X) = 0, and E(X²) = 12/5. Find the pdf of Y = X²

Homework Equations

The Attempt at a Solution

I don't really know where to start.

 

[vela]

There are a couple of methods to do this. One of the easiest to understand conceptually is to calculate the probability that Y is less than or equal to some y in terms of X.

P(Y≤y) = P(X²≤y) = ...

Now P(Y≤y) is just F_Y(y), the cdf of Y, so to find f_Y(y), the pdf of Y, just differentiate it.

 

[Quincy]

so first I should find the cdf of X by integrating the pdf of X, then I should find the cdf of Y, and then differentiate that to get the pdf of Y? How do I get the cdf of Y from the cdf of X^2?

 

[vela]

You don't need to find the cdf of X first, though you certainly can use it to solve the problem. I think you should first determine what interval [a,b] of X corresponds to the condition X²≤y. Then you use P(X²≤y) = P(a≤X≤b). This second probability you can calculate using the cdf of X if you found that or you can find it using f_X(x), whichever you prefer.