Computing the CDF from a PDF

Question

I understand how to find the CDF of Y it would be negative infinity to y for the bounds, but I am stuck since I am given two functions. Any help would be appreciated

Answer 1

Since $\int_{-1}^0\tfrac{-\pi}{8}\sin(\pi y)dy=[\tfrac18\cos\pi y]_{-1}^0=\tfrac14$, $\frac{3}{4c}=\int_0^1\sin(\pi y)dy=\frac{2}{\pi}\implies c=\frac{3\pi}{8}$. So the CDF $F_Y(y)$ increases from $0$ at $y-1$ to $\tfrac14$ at $y=0$ and $1$ at $y=1$, and is given by$$F_{Y}\left(y\right)=\left\{ \begin{array}{rl} 0 & y\le-1\\ \frac{1+\cos\pi y}{8} & y\in\left[-1,\,0\right]\\ \frac{5-3\cos\pi y}{8} & y\in\left[0,\,1\right]\\ 1 & y\ge1 \end{array}\right..$$

Answer 2

You'll have to split up the integral to integrate it. That is, if $-1 \leq y \leq 0$ then $$ F_Y(y) = \int_{-1}^y - \frac{\pi}{8} \sin(\pi y') \ dy' $$ and if $0 < y \leq 1$ then $$ F_Y(y) = \int_{-1}^0 - \frac{\pi}{8} \sin(\pi y') \ dy' + \int_{0}^y c \sin(\pi y') \ dy'. $$