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.
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 $\displaystyle W=X^2$, then the density of W is...
$\displaystyle f_W(w)=f_X(x){1\over 2} w^{-1/2}=e^{-w}$ when w>0 and zero otherwise.
Hence $\displaystyle W\sim Exp(1)=\Gamma(1,1)$ as well as X.
Since they are independent the sum is a $\displaystyle \Gamma(2,1)$, which gives you the density.
The cumulative distribution function is $\displaystyle F_Z(z)=P(Z\le z)=\int_{-\infty}^zf_Z(t)dt$.