# 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! Nov 3, 2017 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. Nov 3, 2017 at 4:33
• The region can be described by $\{ (x,y) : 0 \leq x \le a, 0 \leq y \leq bx \} \cup \{ (x,y) : a \leq x \le 2a, \frac{b}{a}x-b \leq y \leq b \}$ Nov 3, 2017 at 4:57