How to Find the CDF and PDF of (X+Y)²?

[EngWiPy]

Hi,

Suppose we have two random variables X and Y with CDFs and PDFs F_X(x), F_Y(y), f_X(x), and f_Y(y). Now suppose that a new random variable formed as Z=(X+Y)^2. How can we find the CDF F_Z(z) and the PDF f_Z(z) of this new random variable? Note: X and Y are independent random variables.

Thanks

 

[mathman]

If you do it in two steps it is straightforward.
First get the distribution function for W=X+Y (convolution of the distribution functions).
Then get the distribution function for the square of a random variable Z = W².
P(Z < z)=P(-√z < W < √z).

 

[EngWiPy]

If you do it in two steps it is straightforward.
First get the distribution function for W=X+Y (convolution of the distribution functions).
Then get the distribution function for the square of a random variable Z = W².
P(Z < z)=P(-√z < W < √z).

I though in it in another way:

W_1=X^2+Y^2 and W_2=2XY, then Z=W_1+W_2. Your approach seems easier.