Let $~X~$ and $~Y~$ be continuous r.v. with a joint Pdf:
$$ f(x,y)=\begin{cases} 2(x+y) & ; 0\leq x\leq y\leq 1\\ 0 & ; \text{else} \end{cases} $$
Can someone explain how to find the joint CDF of this problem?
Also can someone please explain to me how to evaluate the bounds to integrate cdf and how to create the piecewise of joint cdf.
I’m stuck on this joind cdf and marginal cdf.
$\endgroup$ 51 Answer
$\begingroup$Hint: Firstly we can use that $0\leq x\leq y\leq 1$ is equivalent to $x\leq y\leq 1, 0\leq x\leq 1$
Then $$P(X\leq a, Y\leq b)=\int_0^a \left(\int_x^b 2(x+y) \, dy \right) \, dx $$
$$P(X\leq a, Y\leq b)=\int_0^a \left(b^2+2bx-3x^2\right) \, dx =...$$
$\endgroup$ 1
