Density Function Transformation Probelm

Question

I'm having trouble with this question, the $P(Y \ge y)$ part of the equation is confusing me since its not the cdf and I'm not sure how I should proceed.

Suppose X has density function $x/2$ for $0 < x < 2$ and 0 otherwise. Find the density function of $Y= X(2-X)$ by computing $P(Y \ge y)$ and then differentiating.

Thanks for your help.

Answer 1

Note for any $y\in (0,1)$, $(Y\geq y)=(1-\sqrt{1-y}\leq X\leq 1+\sqrt{1-y})$. Hence $P(Y\geq y)=P(1-\sqrt{1-y}\leq X\leq 1+\sqrt{1-y})=\sqrt{1-y}.$ Therefore, $$f_Y(y)=\left\{\begin{array}{ll}\frac{1}{2\sqrt{1-y}}&\mathrm{if}\,\,0<y<1\\0&\mathrm{otherwise}.\end{array}\right.$$