Independent solution

How to solve this Expected Value question

Setup

Setup

The expectation is the integral of the value multiplied by its density. Because the density contains an absolute value, its algebraic form changes at zero.

E[X]=∫−24x∣x∣10 dxE[X]=\int_{-2}^{4}x\frac{|x|}{10}\,dx

Model

Model

On the negative interval the integrand is negative, while on the positive interval it is positive. Split the integral at zero before evaluating.

E[X]=110(−∫−20x2dx+∫04x2dx)E[X]=\frac1{10}\left(-\int_{-2}^{0}x^2dx+\int_0^4x^2dx\right)

Compute

Compute

The two contributions combine to an expected value of approximately 1.8667.

E[X]=110(−83+643)=2815E[X]=\frac1{10}\left(-\frac83+\frac{64}3\right)=\frac{28}{15}

Answer

Answer

The expected value is approximately 1.8667, selecting choice D.

2815(D)\boxed{\frac{28}{15}\quad\text{(D)}}