Find the first and second moments of a distribution of order statistics?

Question

I'm not totally sure how to even word this question, but I need to find the first and second moments of two variables, $M$ and $N$ as defined by:

$M=\min(X_1,X_2,\dots,X_n)$ and $N=\max(X_1,X_2,\dots,X_n)$ where $(X_1,X_2,\dots,X_n)$ is uniformly distributed on the interval $(0,1)$.

Here's what we have:

$F_N(m)=1-(1-F_X(m))^n$

$f_N(m)=n(1-F_X(m))^{n-1}f_X(m))$

$F_M(m)=F_x(m)^n$

$f_M(m)=nF_X(m)^{n-1}f_X(m)$

I'm a little stuck actually computing the expectations, though. By definition, I know

$E[X^k]=\int_{0}^{1}x^kf_X(x)dx$,

But I'm getting really confused by the actual computation for my specific purposes of calculating $E[M], E[M^2], E[N], E[N^2]$. Thanks for any guidance.

Answer 1

If you were given\taught the Beta function, then, since you know the density of $M$ (the minimum), \begin{align*} E[M] &= \int_0^1 mf_M(m)\,dm\\ &=\int_0^1m\cdot n(1-m)^{n-1}\,dm\\ &=n \int_0^1\cdot m^{2-1}(1-m)^{n-1}\,dm\\ &=n B(2,n)\\ &=n \cdot\frac{(2-1)!(n-1)!}{(2+n-1)!}\\ &=\frac{n!}{(n+1)!}\\ &=\frac{1}{n+1}. \end{align*} Similarly, \begin{align*} E[M^2] &= \int_0^1m^2f_M(m)\,dm\\ &=\int_0^1m^2\cdot n(1-m)^{n-1}\,dm\\ &=n \int_0^1m^{3-1}(1-m)^{n-1}\,dm\\ &=nB(3,n)\\ &=n\cdot\frac{(3-1)!(n-1)!}{(3+n-1)!}\\ &=\frac{2!n!}{(n+2)!}\\ &=\frac{2}{(n+1)(n+2)} \end{align*} You can verify that $$\text{Var}[M] = E[X^2]-\{E[X]\}^2 = \frac{n}{(n+1)^2(n+2)}.$$

This will work $N$ also.

Answer 2

Here I gave you explicitely forms of each distribution.

For $N$:

$\begin{eqnarray}E[N]&=&\int_0^1nx(1-x)^{n-1}dx\\ &=&\frac{n}{n^2+n}\\ &=&\frac{1}{n+1} \end{eqnarray}$

$\begin{eqnarray}E[N^2]&=&\int_0^1nx^2(1-x)^{n-1}dx\\ &=&\frac{2n}{n(n+1)(n+2)}\\ &=&\frac{2}{(n+1)(n+2)} \end{eqnarray}$

Can you do for $M$?

Hint $$E[M]=\int_0^1 {n\cdot x\cdot x^{n-1}} dx\mbox{ and }E[M^2]=\int_0^1 {n\cdot x^2\cdot x^{n-1}} dx$$