Independent solution

How to solve this Mixed-Distribution Expectation question

Setup

Setup

The distribution is mixed: half of its probability is concentrated at zero and the remaining half is spread uniformly over the interval from two to three.

Pr⁡(X=0)=0.5,fX(x)=0.5 for 2<x<3\Pr(X=0)=0.5,\quad f_X(x)=0.5\ \text{for }2<x<3

Model

Model

Compute the expectation by adding the point-mass contribution and the integral of the continuous component.

E[X]=0(0.5)+∫230.5x dxE[X]=0(0.5)+\int_2^3 0.5x\,dx

Compute

Compute

The point mass contributes zero, while the continuous component contributes 1.25.

E[X]=0.5(32−222)=1.25E[X]=0.5\left(\frac{3^2-2^2}{2}\right)=1.25

Answer

Answer

The expected value is 1.25.

1.25(D)\boxed{1.25\quad\text{(D)}}