Independent solution

How to solve this Cumulative Distribution Functions question

Setup

Setup

Differentiate each smooth portion of the cumulative distribution to obtain the density on the support.

f(x)=ddx(x32)=3x22,0<x<1f(x)=\frac{d}{dx}\left(\frac{x^3}{2}\right)=\frac{3x^2}{2},\qquad 0<x<1
f(x)=ddx(2x−x22−1)=2−x,1<x<2f(x)=\frac{d}{dx}\left(2x-\frac{x^2}{2}-1\right)=2-x,\qquad 1<x<2

Model

Model

The CDF is continuous at the join and endpoints, so there are no point masses to add. Compute the first moment from the two density pieces.

E⁡[X]=∫01x3x22 dx+∫12x(2−x) dx\operatorname{E}[X]=\int_0^1 x\frac{3x^2}{2}\,dx+\int_1^2x(2-x)\,dx

Compute

Compute

Evaluate the two contributions separately and combine them.

∫013x32 dx=38\int_0^1\frac{3x^3}{2}\,dx=\frac38
∫12(2x−x2) dx=23\int_1^2(2x-x^2)\,dx=\frac23
E⁡[X]=38+23=2524\operatorname{E}[X]=\frac38+\frac23=\frac{25}{24}

Answer

Answer

The expected value is 25/24.

2524(E)\boxed{\frac{25}{24}\quad\text{(E)}}