Suppose we have $X,Y$, two independent standard normal random variables. How can we calculate
$P(|\min(X,Y)|<1)$.
I am still learning multivariables probability, and I also realize there are a lot of nice properties of two standard normal r.vs but I am not sure how to use them.
Best Answer
For any continuous distribution:
$\Pr(|\min(X,Y)| \lt k) = \Pr(\min(X,Y) \gt -k) - \Pr(\min(X,Y) \ge k)$
$= \Pr(X \gt -k) \Pr(Y \gt -k) - \Pr(X \ge k) \Pr(Y \ge k)$
$ = (1- F(-k))^2- (1- F(k))^2 $.
In the case of a distribution which is symmetric about $0$, this reduces to
$F(k)^2- (1- F(k))^2= 2F(k)-1 = F(k)-F(-k) = \Pr(|X| \le k)$.
which is your result.