Least upper bound for the maximum of two random variables

Question

Two independent random variables are not negative and E[X]=a, E[Y]=b. We define a new random variable Z=max{X,Y}. If t>0 and t>max{a,b} calculate the best (least) upper bound for the probability P[Z>t].

Is it E[Z]=a*b?

Answer 1

You know that the variables are independent, which means $$P(Z\leq t)=P(X\leq t \wedge Y\leq t)=P(X\leq t)P(Z\leq t),$$ or equivalently $$P(Z>t)=1-(1-P(X>t))(1-P(Y>t)).$$ So use Markov's inequality to get bounds on $P(X>t)$ and $P(Y>t)$ and combine these to get a bound on $P(Z>t)$. This will be the best you can do, since it is possible for Markov's inequality to be tight for both $X$ and $Y$ (for a specific value of $t$).

Answer 2

The answer is $P(Z>t) < \frac{at+bt-ab}{t^2}$