Question: The quantity $2 - 2Y^2$ defines a random variable $W$. $Y$ follows a gamma distribution with $\alpha = 3$ and $\lambda = 0.5$.
Find the density function for $W$.
This is a problem I came across during a review session for my probability final. Given that $Y$ follows a gamma distribution, we can find the density function for $Y$ fairly easily. However, I'm getting stuck after that point. How can you find the density function of $W$ given the density function of $Y$ when you know how $Y$ and $W$ are related to each other?
$P(2-2Y^{2} \leq x)=P(Y^{2} \geq \frac {2-x} 2)$. This is $0$ if $x \geq 2$ For $x <2$ we get $P(2-2Y^{2} \leq x)=P(Y\geq \sqrt {1-\frac x 2})=1-P(Y\leq \sqrt {1-\frac x 2})$. Write down the value of this using Gamma density and differentiate w.r.t. $x$. That gives you the density of $W$.