Given the joint pdf: $$g(x,y) = \begin{Bmatrix} Bxy &\text{if } x+y \leq 1 \\ 0& \text{o.w.} \end{Bmatrix}$$
I want to know the value of $B$ for $g(x,y)$ to be a proper pdf.
My attempt: We need to have $g(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} Bxy = 1$ for a proper pdf.
So $$g(x,y) = \int_{0}^{1} \int_{0}^{1} Bxy = 1$$
From here, I can solve for $B$, which equals $4$.
Is my solution correct? While I am sure about the range of values for $x$, I am not sure about the range of $y$.
Edit: From the hint below, I found $B=24$ for $g$ to be a proper pdf. However, I am not sure how to find the expectations $E(X)$ and $E(Y)$. Any help with that would be appreciated.
I am using these formulas and ranges $E(X) = \int_{0}^{1}x g(x,y) dx$ $E(Y) = \int_{0}^{1}y g(x,y) dy$ to find both
First of all you would want to restrict $x$ and $y$ to positive values. Otherwise $g$ is not integrable so there is no value of $B$ to make it a density function.
Your answer is not correct even after this modification . You need $\int _0^{1}\int_0^{1-y} Bxydxdy=1$. This gives $\int_0^{1} By(1-y)^{2}/2dy=1$. I will let you finish.
$EX=\int _0^{1}\int_0^{1-y} Bx^{2}ydxdy$ and $EY=\int _0^{1}\int_0^{1-y} Bxy^{2}dxdy$.