# Math Help - Find the CDF for Z

1. ## Find the CDF for Z

Suppose X and Y are independent random variables, where Y is exponentially distributed with mean 1, and X has density function
fX(x) = 2x*exp{-x^2}
(x >= 0):
Find the CDF for Z = X^2 + Y .

I don't know what is CDF stand for.

2. That's easy.
I can give you the density instead of the CDF, the cdf is easy too.
http://en.wikipedia.org/wiki/Cumulat...ution_function

Let $W=X^2$, then the density of W is...

$f_W(w)=f_X(x){1\over 2} w^{-1/2}=e^{-w}$ when w>0 and zero otherwise.

Hence $W\sim Exp(1)=\Gamma(1,1)$ as well as X.

Since they are independent the sum is a $\Gamma(2,1)$, which gives you the density.

The cumulative distribution function is $F_Z(z)=P(Z\le z)=\int_{-\infty}^zf_Z(t)dt$.