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Let $X_{1},X_{2},...,X_{n}$, for $n=2,3,...,$ be independent and identically distributed random variables with common distribution function. $F_{X}$ . Find a formula for the joint distribution $F_{Y,Z}$ for

$$Y=\max\{X_{1}, X_{2},...,X_{n}\} \quad \text{and} \quad Z=\min\{X_{1},X_{2},...,X_{n}\}$$

in terms of $F_{X}$.

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Do you know how to compute the marginal distribution of the maximum and the minimum ? – leonbloy May 1 '12 at 16:07
As was already mentioned about another post by the same OP, with no answer from said OP: In order to help you with homework, you need to show us what you've done so far. – Did May 1 '12 at 16:21
Please improve the title! – Rasmus May 1 '12 at 17:54
up vote 3 down vote accepted

We have $$\begin{align*} F_{Y, Z}(y, z) &= \mathbb{P}(Y \leq y \ \text{and} \ Z \leq z)\\ &= \mathbb{P}(Y \leq y) - \mathbb{P}(Y \leq y \ \text{and} \ Z > z)\\ &= \mathbb{P}(\forall i, \ X_i \leq y) - \mathbb{P}(\forall i, \ z < X_i \leq y)\\ &\stackrel{(1)}{=} \mathbb{P}(X \leq y)^{n} - \mathbb{P}(z < X \leq y)^{n}\\ &\stackrel{(2)}{=} F_X(y)^{n} - (F_X(y) - F_X(z))^{n}, \end{align*}$$ where we have used independence assumption at (1), and identical distribution assumption at (2).

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