$X$ 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.
$X$ 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.
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.