Independent solution

How to solve this Order Statistics question

Setup

Setup

Let M be the largest of the three independent uniform observations.

M=max⁡(X1,X2,X3),Xi∼Uniform⁡(0,10)M=\max(X_1,X_2,X_3),\qquad X_i\sim\operatorname{Uniform}(0,10)

Model

Model

The maximum does not exceed x exactly when all three observations do not exceed x.

FM(x)=Pr⁡(X1≤x,X2≤x,X3≤x)=(x10)3F_M(x)=\Pr(X_1\le x,X_2\le x,X_3\le x)=\left(\frac{x}{10}\right)^3
fM(x)=3x2103,0≤x≤10f_M(x)=\frac{3x^2}{10^3},\qquad 0\le x\le10

Compute

Compute

Integrate x against the maximum density.

E⁡[M]=∫010x3x2103 dx\operatorname{E}[M]=\int_0^{10}x\frac{3x^2}{10^3}\,dx
E⁡[M]=31000[x44]010=304=7.5\operatorname{E}[M]=\frac{3}{1000}\left[\frac{x^4}{4}\right]_0^{10}=\frac{30}{4}=7.5

Answer

Answer

The expected largest observation is 7.5.

7.5(B)\boxed{7.5\quad\text{(B)}}