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.