# Homework Help: Non 1-1 transformation of continuous random variable

1. Feb 2, 2010

### Kate2010

1. The problem statement, all variables and given/known data

X is exponentially distributed with mean s.
Find P(Sin(X)> 1/2)

2. Relevant equations

fX(x) = se-sx, x$$\geq$$ 0
0, otherwise

FX(x) = 1 - e-sx, x$$\geq$$ 0
0 otherwise

3. The attempt at a solution

Let Y = sin X

FY (y) = P(Y$$\leq$$ y)
= P(sinX $$\leq$$ Y)
= P(X $$\leq$$ arcsin(y), X$$\geq$$ $$\pi$$ - arcsin(y)) {This is where I become slightly unsure}
=FX(arcsin(y)) - FX($$\pi$$ - arcsin(y))
=1-e-s(arcsin(y)) - (1-e-s($$\pi$$ - arcsin(y)))

From here I can differentiate to find the pdf and then use that to find sinX < 1/2.

2. Feb 4, 2010

### Kate2010

Any ideas anyone?