Independent solution

How to solve this Order Statistics question

Setup

Setup

Find the population cumulative distribution on the unit interval.

F(x)=∫0x2t dt=x2F(x)=\int_0^x2t\,dt=x^2

Model

Model

Use the density of the second order statistic from a sample of three.

g(y)=6F(y)[1−F(y)]f(y)g(y)=6F(y)[1-F(y)]f(y)
g(y)=12y3(1−y2),0<y<1g(y)=12y^3(1-y^2),\qquad 0<y<1

Compute

Compute

Calculate the first two moments and subtract the squared mean.

E⁡[Y]=12∫01(y4−y6) dy=2435\operatorname{E}[Y]=12\int_0^1(y^4-y^6)\,dy=\frac{24}{35}
E⁡[Y2]=12∫01(y5−y7) dy=12\operatorname{E}[Y^2]=12\int_0^1(y^5-y^7)\,dy=\frac12
Var⁡(Y)=12−(2435)2=732450=0.0297959…\operatorname{Var}(Y)=\frac12-\left(\frac{24}{35}\right)^2=\frac{73}{2450}=0.0297959\ldots

Answer

Answer

The variance of the sample median is approximately 0.030.

0.030(B)\boxed{0.030\quad\text{(B)}}