# joint pdf of two random variables

A pair of random variable $(X, Y)$ is uniformly distributed in the quadrilateral region with $(0,0),(a,0),(a,b),(2a,b)$, where $a,b$ are positive real numbers.

What is the joint pdf $f(X,Y.)$

Find the marginal probability density functions $f_X (x)$ and $f_Y (y)$.

Find $E(X)$and $E(Y)$.

Find $E(X,Y)$.

My understanding is the uniformly distributed pdf $=\frac1{\text{area}}=\frac1{ab}$. It seems that $X$ and $Y$ are independent R.V. because it seems the joint pdf can be factored as $\frac1a \frac1b$ if the pdf is correct. However, after I found the marginal pdf $f_X=\frac2{a^2}; f_Y=\frac2{b}-\frac{2}{b^2}$, which shows $X, Y$ are not independent. If I can not get correct marginal pdf, I can not finish the question d and e. Could anyone help me out? Thank you!

• $X$ and $Y$ are not independent, and the expressions of $f_X(x)$ and $f_Y(y)$ are incorrect! – Math Lover Nov 3 '17 at 3:59

Recheck your value value for $f_Y$ as well. You should get a simpler expression. Notice that pdf should integrate to $1$ of which both of your proposed density functions have failed the test.
• you should state the support of the pdf, that is the region where the density is positive. $\frac1{ab}$ is fine but over which region. – Siong Thye Goh Nov 3 '17 at 4:33
• The region can be described by $\{ (x,y) : 0 \leq x \le a, 0 \leq y \leq bx \} \cup \{ (x,y) : a \leq x \le 2a, \frac{b}{a}x-b \leq y \leq b \}$ – Siong Thye Goh Nov 3 '17 at 4:57