Finding the covariance when given joint pdf

Question

If the joint probability density of X and Y given by

$$ f(x,y) = \begin{cases} 1/4(2x+y), & 0<x<1, 0<y<1\\ 0, & \text{elsewhere} \end{cases} $$

Can I get some hints on how to find the Cov(x,y)?

Answer 1

Outline: We want to calculate $E(XY)-E(X)E(Y)$. The three expectations can each be found by evaluating the appropriate double integral.

Answer 2

Maybe we could just consider the question through the relationship between marginal distribution and joint distribution.

The marginal probability density of y is $f_Y(y)=\int_0^1f(x,y)dx$=1/4(1+y),

so $E(Y)=\int_0^1yf_Y(y)dy=5/24$

then just calculate $E(X)$ and $E(XY)=\int_0^1\int_0^1xyf(x,y)dydx$.