Are max and min of n iid r.v.s. independent?

[maverick280857]

Hi

Suppose $X_{1}, \ldots, X_{n}$ is a sequence of i.i.d. random variables. We define

$$X_{(n)} = max(X_{1}, \ldots, X_{n})$$
$$X_{(1)} = min(X_{1}, \ldots, X_{n})$$

Are $X_{(n)}$ and $X_{(1)}$ independent?

Whats the best/easiest way to verify this?

Thanks
Vivek

 

[Pere Callahan]

They are not independent. The maximum is always larger than the minimum ...

 

[maverick280857]

Yeah, nice observation. Thanks

 

[maverick280857]

Suppose I wanted to show it using the factorization of the joint pdf or joint pmf, how would I do that?

 

[Pere Callahan]

You just have to find one example such that the cdf does not factorize.

Let m be the minimum, M the maximum, x some real number

What about P(m<x && M<x)

This is equal to only M being less than x ( because then m is automatically also less than x.

so P(m<x && M<x) = P(M<x)

For this to be equal to the factorized probability P(m<x)P(M<x) you need to have P(m<x)=1 for all real x ...which is not true

 

[maverick280857]

Thanks Pere

 

[statdad]

Look for a statistics text that discusses the distributions of order statistics and sets of order statistics. You will be able to find a general formula for the p.d.f of the $$\min \text{ and } \max$$ in terms of the marginal pdfs and joint pdf of the sample. Once you see that form, you will see that they need not be independent.