Given $X\sim \exp(\lambda)$ and $Y = X^3$, what would be their joint cumulative distribution, $F(x, y)$? Since $X$ and $Y$ are dependent, I can't just integrate the product of their probability density functions: $f(x)f(y)$. I've got it down to:
$\begin{align*} F(x,y) &= P(X \le x, Y \le y) \\ &= P(X \le x, X^3 \le y) \\ &= P(X \le x, X \le y^{1/3}) \\ &= P(X \le \min(x, y^{1/3})) \end{align*}$
but don't know how to proceed from here on. I've determined the probability density functions for f(x) and f(y). If they were independent, I would just integrate f(y)f(x) dx dy Since they are not independent, I'm not sure how to derive their joint probability distribution. Once I have that, my assumption is that I can just integrate that to derive their joint cdf.