Independent solution

How to solve this Uniform Distribution question

Answer in brief

For a uniform interval [a,b], the median condition gives a+b=12 and the 90th-percentile condition gives a+0.9(b-a)=13.2. Thus a=-3 and b=15, yielding second moment 63 and choice E.

Setup

Setup

Let a and b be the lower and upper endpoints. Uniform percentiles advance linearly across the interval.

qp=a+p(b−a)q_p=a+p(b-a)

Model

Model

Translate the median and 90th-percentile information into two endpoint equations.

a+b2=6\frac{a+b}{2}=6
a+0.9(b−a)=13.2a+0.9(b-a)=13.2

Compute

Compute

Solve the two equations, then use the raw second-moment formula for a continuous uniform distribution.

a=−3,b=15a=-3,\qquad b=15
E⁡[X2]=a2+ab+b23\operatorname{E}[X^2]=\frac{a^2+ab+b^2}{3}
E⁡[X2]=9−45+2253=63\operatorname{E}[X^2]=\frac{9-45+225}{3}=63

Answer

Answer

The second raw moment equals 63.

63(E)\boxed{63\quad\text{(E)}}