# order statistics with Bernoulli variables

$X$ 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.

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Hint: $(X,Y)$ can have four different values. Make a table with four rows and five columns. On each row, write the values of $X$, $Y$,the probability that $(X,Y)$ has these values (sub-hint: iid Bernoulli$(p)$ might help), $\max(X,Y)$, and $\min(X,Y)$. Stare very hard at the array that you have created. – Dilip Sarwate Aug 3 '12 at 17:54
I do not know what you call the formula for marginal order statistics pmf but you could simply enumerate the values of $(i,j)$ such that the event $[\max(X,Y)=i,\min(X,Y)=j]$ is not empty and compute its probability. – Did Aug 3 '12 at 17:56

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.