Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $F_{X}$.

share|improve this question
    
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
1  
Please improve the title! –  Rasmus May 1 '12 at 17:54
add comment

1 Answer 1

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.