I have this joint probability question I'm having trouble with. If X, Y follow a poisson distribution with Poisson($\lambda$) and X and Y are independent. Show that:
$$1 \le \frac{VAR(max{X,Y})}{\lambda} \le 2$$
I know that variance of a random variable following a poisson distribution is $\lambda$. Any help would be greatly appreciated!
Best Answer
In fact, it is always true that for two random variables $X$ and $Y$ (independent or not), $\text{Var}(\max(X,Y)) + \text{Var}(\min(X,Y)) \le \text{Var}(X) + \text{Var}(Y)$. This gives you the upper bound. I don't see an easy way to get the lower bound.