Joint CDF to Marginal CDF - For Continuous R.V. X & Y

Recently I had a question on an exam, that had me finding the Marginal CDF with only the Joint CDF available. The question was exactly:

$$F_{X,Y}(x,y) = \begin{cases} (1-e^{-\alpha x})(1-e^{- \beta y}) & x \ge0, y \ge0 \\ 0 & \text{otherwise} \end{cases}$$

I had to find $F_X(x)$ and $F_Y(y)$, frankly, I didn't know how to do it.

I got to the point of getting $f_X(x)$ and $f_Y(y)$ but got stuck there.

Solution 1:

When you have $F(x,y)$, you get: $$P(X\le x)=P(X\le x, Y<+\infty)=\lim_{y\to\infty}F(x,y)$$

$$P(Y\le y)=P(X<+\infty,Y\le y)=\lim_{x\to\infty}F(x,y)$$

So you can compute $F_X$ and $F_Y$ passing to the limit.