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.
Best Answer
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 $f(x)$:
Then, the joint pdf of the $r^{\text{th}}$ and $s^{\text{th}}$ order statistics is say $g(x_r, x_s)$:
where I am using the
OrderStatfunction 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:
All done. This should hopefully help in both forming the appropriate integrals, and checking your working.