Can someone please help me out with this sum? It says:

If $(X_1, X_2, \dots, X_n)$ are a random sample from Uniform(0,1) distribution, find the correlation between the order statistics $X$ order $r$ and $X$ order $s$, $r < s$.

I have used the formula for correlation coefficient as:

$$\rho=\frac{E[X_{(r)}X_{(s)}]-E[X_{(r)}]E[X_{(s)}]}{\sqrt{Var(X_{(r)}) Var(X_{(s)})}}$$

I know the order statistic follows a beta distribution and hence found out the mean and variance of $X_{(r)}$ and $X_{(s)}$. The problem is, I'm unable to compute $E[X_{(r)}X_{(s)}]$. I have tried to find the joint expectation by finding the joint probability distribution of the rth and sth order statistics, but when I try to multiply $X_{(r)}$ and $X_{(s)}$ with the joint PDF and integrate it to find the joint expectation, I am unable to integrate it.

  • $\begingroup$ To find $E[X_{(r)}X_{(s)}]$, you will need the joint pdf of $(X_{(r)},X_{(s)})$. Have you derived this? $\endgroup$
    – wolfies
    May 24 '16 at 17:08
  • $\begingroup$ @wolfies yes, I have. Then I tried to find the E(X(r)X(s)) by integrating the product of the joint distribution and X(r)X(s) dx(r)dx(s) over (0,1). But I couldn't integrate the product $\endgroup$
    – Sweta95
    May 24 '16 at 17:13
  • $\begingroup$ math.stackexchange.com/q/4104115/321264 $\endgroup$ Apr 18 at 11:06

You appear to understand the steps involved. I am not sure if this is an assignment or exercise, so it may not be appropriate to show workings anyway, but am happy to sketch out the approach ...

Given: $X \sim \text{Uniform}(0,1)$ with pdf $f(x)$:

enter image description here

Then, the joint pdf of the $r^{\text{th}}$ and $s^{\text{th}}$ order statistics is say $g(x_r, x_s)$:

enter image description here

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the calculation.

Given the joint pdf of the $r^{\text{th}}$ and $s^{\text{th}}$ order statistics, you seek their correlation:

enter image description here

All done. This should hopefully help in both forming the appropriate integrals, and checking your working.

  • $\begingroup$ It isn't an assignment, just a practice problem. My approach was exactly similar to yours,sir. Maybe I should check the integration again. Thank you. :) $\endgroup$
    – Sweta95
    May 24 '16 at 17:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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