Independent solution

How to solve this Continuous Distributions question

Setup

Setup

Shift the variable to the midpoint of its support by setting U=X-6.

−1<U<1-1<U<1
fU(u)=32u2f_U(u)=\frac{3}{2}u^2

Model

Model

The centered density is even, so U has mean zero. Translation does not change variance.

E⁡[U]=0\operatorname{E}[U]=0
Var⁡(X)=Var⁡(U)=E⁡[U2]\operatorname{Var}(X)=\operatorname{Var}(U)=\operatorname{E}[U^2]

Compute

Compute

Integrate the centered second moment over the two-sided unit interval.

E⁡[U2]=∫−11u2(32u2) du\operatorname{E}[U^2]=\int_{-1}^{1}u^2\left(\frac{3}{2}u^2\right)\,du
E⁡[U2]=32[u55]−11=35=0.60\operatorname{E}[U^2]=\frac{3}{2}\left[\frac{u^5}{5}\right]_{-1}^{1}=\frac{3}{5}=0.60

Answer

Answer

The variance is 0.60 square units.

0.60(B)\boxed{0.60\quad\text{(B)}}