# Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ .

I got this problem where:

The random variables $$X_1, X_2,$$ and $$X_3$$ are independent and $$Exp(1)-$$ distributed.
Compute the correlation coefficient $$r(X_{(1)},X_{(3)})$$ .

I know through research that: $$\mathrm E(X_{(k)})=\sum\limits_{i=n-k+1}^n\frac1i,\qquad \mbox{Var}(X_{(k)})=\sum\limits_{i=n-k+1}^n\frac1{i^2}.$$

I could find everything except, $$E(X_{(1)}X_{(3)})$$.

I got: $$E(X_{(1)})=\frac {1}{3},\qquad E(X_{(3)})=1+\frac {1}{2}+\frac {1}{3}$$

$$Var(X_{(1)})= \frac {1}{9}, \qquad Var(X_{(3)})= 1+\frac {1}{4}+\frac {1}{9}$$

I also know that correlation = $$\frac{cov(X_{(1)},X_{(3)})}{\sqrt{Var(X_{(1)})Var(X_{(3)})}}$$

Update:

So from the implication of @NcH I figured the $$E(X_{(1)}X_{(3)})=1$$ But, when trying to compute the correlation coefficient I get 1 as the final answer as opposed to 2/7 suggested by the textbook.

$$r= \frac{1-11/18}{7/18}=1$$

Anyone has a different answer ?

• Your solutions assume that the sample size is 3, but this not stated in the question. The answer should therefore be a function of the sample size $n$. – wolfies Feb 21 at 12:21
• Side note, the answer is 2/7 – Mahamad A. Kanouté Feb 21 at 12:23
• @wolfies it is explicitly stated in the question, while OP used the results which are true for sample size $n$ and $k$-th order statistics. – pointguard0 Feb 21 at 12:35
• @pointguard Why would the expectation of the sample minimum in a sample of size $n$ be equal to $\frac13$? – wolfies Feb 21 at 12:40
• it is not equal to $\frac 1 3$, it is equal to $\sum_{i=n-k+1}^n \frac 1 i$. – pointguard0 Feb 21 at 12:41

I think that the simplest way is to use joint PDF of $$X_{(1)}$$ and $$X_{(3)}$$. For $$0 using formula for [joint pdf of two order ststistics]1 we get $$f_{X_{(1)}, X_{(3)}}(x,y) = 3! \bigl((1-e^{-y})-(1-e^{-x})\bigr)e^{-x}e^{-y}=6(e^{-2x-y}-e^{-x-2y})$$
Then $$\mathbb E[X_{(1)}X_{(3)}]=\iint_{0
• Thanks, I added lost $6$. No, this integral is not equal to $1$. Wolframalpha gives answer $\frac{13}{18}$. – NCh Feb 22 at 2:42
• x goes from 0 to y and y goes from 0 to $\infty$ right ? – Mahamad A. Kanouté Feb 22 at 2:45