Independent solution

How to solve this Continuous Random Variables question

Setup

Setup

Split each moment at zero because the absolute-value density changes sign in its algebraic expression there.

f(x)={−x/10,−2<x<0,x/10,0≤x<4.f(x)=\begin{cases}-x/10,&-2<x<0,\\x/10,&0\le x<4.\end{cases}

Model

Model

Calculate the first and second raw moments from the two density branches.

E[X]=∫−20x−x10 dx+∫04xx10 dxE[X]=\int_{-2}^{0}x\frac{-x}{10}\,dx+\int_{0}^{4}x\frac{x}{10}\,dx
E[X2]=∫−20x2−x10 dx+∫04x2x10 dxE[X^2]=\int_{-2}^{0}x^2\frac{-x}{10}\,dx+\int_{0}^{4}x^2\frac{x}{10}\,dx

Compute

Compute

Evaluate both moments and apply the raw-moment variance identity.

E[X]=2815,E[X2]=345E[X]=\frac{28}{15},\qquad E[X^2]=\frac{34}{5}
Var⁡(X)=345−(2815)2\operatorname{Var}(X)=\frac{34}{5}-\left(\frac{28}{15}\right)^2
Var⁡(X)=746225=3.315555…\operatorname{Var}(X)=\frac{746}{225}=3.315555\ldots

Answer

Answer

The variance rounds to 3.32.

3.32(C)\boxed{3.32\quad\text{(C)}}