Given random variable $X\sim \mathrm{Exp}(λ)$, what's the CDF of $Y=\cos(\pi X)$?

Question

Given random variable $X\sim \mathrm{Exp}(\lambda)$, what's the CDF of $Y=\cos(\pi X)$ ?

I know that the steps should be $F(Y)=P(Y\leq y)=P(\cos(\pi X)\leq Y)$, but the following steps are a bit confusing.

Answer 1

Hint

The idea is to write down the probability on the RHS. So you are almost right: $$ F_Y(y) = \mathbb{P}[Y \le y] = \mathbb{P}[\cos(\pi X) \le y], \quad \forall y \in \mathbb{R} $$ which will need to be converted to an inequality in $X$ and then evaluated using the fact that $X \sim \mathcal{Exp}(\lambda)$.

Think about:

1. What if $y \ge 1$?
2. What if $y \le -1$?
3. For $-1 < y < 1$, you can think of using, e.g. the $\arccos$ to invert the inequality.