# Minimum CDF of random variables

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

$F_{X,Y}(x,y)=P(X≤x,Y≤y)$.

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

I want to know both cases. i.e. if $X,Y$ are not-independent and independent

Regards

Let $x$ by any number. Consider the event $\min(X,Y)\le x$. It can be expressed as the union of two events

shown by the overlapping yellow and green regions in this figure, respectively: The intersection of these events (shown in the bottom left corner where they overlap) obviously is $\{X\le x,\,Y\le x\}=\max(X,Y)\le x$. Therefore (by the PIE),

All three probabilities are given directly by $F$ (answering the main question):

The use of "$\infty$" as an argument refers to the limit; thus, e.g., $F_X(x)=F_{X,Y}(x,\infty)=\lim_{y\to\infty} F_{X,Y}(x,y).$

The result can be expressed in terms of the marginal distributions (only) when $X$ and $Y$ are independent, for then $(1)$ becomes

The latter expression is recognizable as computing the chance that independent variables $X$ and $Y$ are both not less than or equal to $x$, given by $(1-F_X(x))(1-F_Y(x))$: the subtraction from $1$ then gives the complementary chance that at least one of those variables is less than or equal to $x$, which is precisely what $\min(X,Y)\le x$ means. Thus $(1)$ is the natural generalization of $(2)$ to all bivariate distributions.

As a final comment, please note that care is needed in the use of "$\le$" and "$\lt$". They can be interchanged in all the preceding calculations when $F$ is continuous, but otherwise they make a difference.