order statistics with Bernoulli variables

Question

$X$ and $Y$ are iid Bernoulli($p$), then what's the marginal pmf and joint pmf for $\max(X, Y)$ and $\min(X, Y)$?

Not sure if I can use the formula for marginal order statistics pmf here.

Answer 1

Let $U=\min(X,Y)$, $V =\max(X,Y)$. Since $U \leqslant V$, possible values of the pair $(U,V)$ are $(0,0)$, $(0,1)$ and $(1,1)$. You now have to compute these probabilities: $$ \mathbb{P}(U=0,V=0) \stackrel{\text{why?}}{=} \mathbb{P}(X=0,Y=0) = \underline{\phantom{1-p}}^2 $$ $$ \mathbb{P}(U=1,V=1) \stackrel{\text{why?}}{=} \mathbb{P}(X=1,Y=1) = \underline{\phantom{p}}^2 $$ $$ \mathbb{P}(U=0,V=1) \stackrel{\text{why?}}{=} 1 - \mathbb{P}(U=0,V=0) - \mathbb{P}(U=1,V=1) $$ To compute marginals, apply the definition, e.g.: $$ \mathbb{P}(U=0) = \mathbb{P}(U=0,V=0) + \mathbb{P}(U=0,V=1) $$ etc.