# Function of random variables

1. Jul 25, 2014

### estro

Suppose X ~ U[ 0, pi ]

What is the distribution of Y=sinX.

I have a solution in my notes however I don,t understand the following the second transition:
$$F_Y(y) = P(Y \leq y) = P(X \leq \arcsin(y)) + P(X \geq \pi - \arcsin(y)) = ...$$

Where the $P(X \geq \pi - \arcsin(y))$ comes from?

2. Jul 25, 2014

### Orodruin

Staff Emeritus
The sine function is not monotonic.

Y is smaller than y if either of the arguments on the RHS are fulfilled.

3. Jul 25, 2014

### Infinitum

Since the sin function is symmetric about ∏/2, there are two possible ranges you need to consider for the same values(imagine drawing a straight horizontal line at any 'y' on the sin curve, there are two symmetric value intervals below it), from [0, arcsin(y)] and [∏ - arcsin(y), ∏] which can be reflected in the solution.