# Continuous random variable - transformation using sin

1. Feb 5, 2010

### Kate2010

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

There is a pin of length 4 which appear on a photograph, and the length of the image observed is y, an observation on the random variable Y. The pin is at an angle x, 0$$\leq$$x$$\leq$$$$\pi$$/2, to the normal to the film, this is an observation on the r.v. X.

1. If all angles X are equally likely then derive the distribution of Y.

2. What is E(Y)?

2. Relevant equations

3. The attempt at a solution

X is uniform so fX(x) = 2/$$\pi$$ , 0$$\leq$$x$$\leq$$$$\pi$$/2 and fX (x) = 0 otherwise.

So FX (x) = 0, x<0
= 2x/$$\pi$$, 0$$\leq$$x$$\leq$$$$\pi$$/2
=1, x>$$\pi$$/2

Y=4SinX

FY(y)= P(Y$$\leq$$y)
=P(4sinX $$\leq$$y)
=P($$\pi$$ - arcsin(y/4) $$\leq$$ X $$\leq$$ 2$$\pi$$ - arcsin(y/4)) (*)
= FX (2$$\pi$$ - arcsin(y/4)) - FX ($$\pi$$ - arcsin(y/4)
= 2 + (2/$$\pi$$)arcsin(y/4)

So fY(y) = (2/$$\pi$$)(1/4)(1/$$\sqrt{1-y2 /4}$$) for 0$$\leq$$y$$\leq$$4 and 0 otherwise.

Is this ok so far? I'm very unsure of the stage marked by (*).

Last edited: Feb 5, 2010