There is the following question:
Let $X_{(1)},X_{(2)},X_{(3)}$ be statistic order of three independent random variables $X_1,X_2,X_3$ with uniform distribution in $[0,1]$. Find the Correlation coefficient between $X_{(1)},X_{(3)}$.
We know that $X_{(k)}\sim Beta(k,4-k)$ so we get:
$$
Var\left(X_{(k)}\right)=\frac{k\cdot(4-k)}{(k+(4-k))^{2}\cdot(k+(4-k)+1)}=\frac{k(4-k)}{80}, E\left(X_{(k)}\right)=\frac{k}{(4-k)+k}=\frac{k}{3}
$$
We can use the following theorem to calculate $Corr\left(X_{(1)},X_{(3)}\right)$:
$$
Corr\left(X_{(1)},X_{(3)}\right)=\frac{Cov\left(X_{(1)},X_{(3)}\right)}{\sqrt{Var\left(X_{(1)}\right)}\sqrt{Var\left(X_{(2)}\right)}}=\frac{E\left(X_{(1)},X_{(3)}\right)-E\left(X_{(1)}\right)E\left(X_{(3)}\right)}{\sqrt{Var\left(X_{(1)}\right)}\sqrt{Var\left(X_{(2)}\right)}}
$$
The only thing left to calculate is $E\left(X_{(1)},X_{(3)}\right)$. In the solution it says that the Probability density functions are:
I'm not understanding how they calculated the left function. Will be glad to see some explanation. Which theorem did they use?
Best Answer
The tripple joint density function for the order statisics is the probability denisity function for arrangments of the samples that fits those three ordered values, $x\leqslant y\leqslant z$.
Since these three samples are identically and independently distributed, that is:
$$\begin{align}f_{\small\! X_{(1)},X_{(2)},X_{(3)}}\!(x,y,z) &={( f_{\small\! X_1,X_2,X_3\!}(x,y,z) + f_{\small\! X_1,X_2,X_3\!}(x,z,y)+f_{\small\! X_1,X_2,X_3\!}(y,x,z)\\+f_{\small\! X_1,X_2,X_3\!}(y,z,x)+f_{\small\! X_1,X_2,X_3\!}(z,x,y)+f_{\small\! X_1,X_2,X_3\!}(z,y,x))~\mathbf 1_{x\leqslant y\leqslant z}} \\[1ex] &= 3!\,f_{\!\small X_1}\!(x)\,f_{\!\small X_1}\!(y)\,f_{\!\small X_1}\!(z))\;\mathbf 1_{x\leqslant y\leqslant z}\\[1ex]&=3!\,\mathbf 1_{0\leqslant x\leqslant y\leqslant z\leqslant 1}\end{align}$$
The marginal for the joint pdf for $X_{(1)}$ and $X_{(3)}$ is just the integral of this for all middle values between the least and most order statistic.
$$\begin{align}f_{\small\! X_{(1)},X_{(3)}}\!(x,z) &=\int_x^z f_{\small\! X_{(1)},X_{(2)},X_{(3)}}\!(x,y,z) ~\mathrm d y \\[2ex]&= 3!~(z-x)~\mathbf 1_{0\leqslant x\leqslant z\leqslant 1}\end{align}$$
Similarly: $$\begin{align}f_{\small X_{(1)}}(x)&= 3\,(1-x)^2~\mathbf 1_{0\leqslant x\leqslant 1}\\[3ex]f_{\small X_{(2)}}(y)&=3!\,y(1-y)\,\mathbf 1_{0\leqslant y\leqslant 1}\\[3ex]f_{\small X_{(3)}}(z)&= 3\,z^2\,\mathbf 1_{0\leqslant z\leqslant 1}\end{align}$$
That is all.