Joint CDF of independent random variables

Question

I know that The joint cumulative function of two random variables X and Y is defined as:

$F_{XY}(x,y)=P(X≤x,Y≤y)$.

How can I find the CDF is $x=y$. In other words is what will be $Pr\{min(X,Y)<x\}$ where $x=y$.

If I already know the individual CDF of both $X$ and $Y$ i.e. $F_{X}(x)$ and $F_{Y}(x)$, can they be useful to compute the $Pr\{min(X,Y)<x\}$?

Any kind of help will be very much appreciated.

Regards

Answer 1

We have \begin{align} \Pr(\min(X, Y) \leq x) &= \Pr(X \leq x \ \lor\ Y \leq x) \\ &= \Pr(X \leq x) + \Pr(Y \leq x) - \Pr(X \leq x \ \land\ Y \leq x) \\ &= F_X(x) + F_Y(x) - F_{X,Y}(x, x) \end{align}