What is the Expected Value of Two Random Variables with a Joint PDF?

[rzn972]

Homework Statement

two random variables have a join pdf
f(x,y)= c , x^2+y^2<= 1, x >0, y>0
0, otherwise
Find c.
Find E{xy}.

Homework Equations

The Attempt at a Solution

For the first part, since area= pi/4,
c(pi/4) = 1
c=(4/pi)
b)
x=cos θ
y= sin θ
∫∫-(sin θ)^2(cos θ)^2 dθ dθ = -308. (integral from 0 to pi/2 for both bounds)
This doesn't make sense to me. x,y are always positive how is the expected value negative? Am I doing something wrong?

 

[LCKurtz]

Homework Statement

two random variables have a join pdf
f(x,y)= c , x^2+y^2<= 1, x >0, y>0
0, otherwise
Find c.
Find E{xy}.

Homework Equations

The Attempt at a Solution

For the first part, since area= pi/4,
c(pi/4) = 1
c=(4/pi)
b)
x=cos θ
y= sin θ
∫∫-(sin θ)^2(cos θ)^2 dθ dθ = -308. (integral from 0 to pi/2 for both bounds)
This doesn't make sense to me. x,y are always positive how is the expected value negative? Am I doing something wrong?

You need $x=r\cos\theta,~y=r\sin\theta,~dydx=rdrd\theta$. And where did that minus sign come from anyway?