Independent solution

How to solve this Continuous Distributions question

Setup

Setup

A positive proportionality constant does not affect where the density is largest.

h(x)=xe−x2,0<x<1h(x)=xe^{-x^2},\qquad 0<x<1

Model

Model

Differentiate the kernel using the product and chain rules.

h′(x)=e−x2−2x2e−x2h'(x)=e^{-x^2}-2x^2e^{-x^2}
h′(x)=e−x2(1−2x2)h'(x)=e^{-x^2}(1-2x^2)

Compute

Compute

Solve the interior stationary equation and verify the derivative changes from positive to negative.

1−2x2=0  ⟺  x=121-2x^2=0\iff x=\frac1{\sqrt2}
12=0.7071068\frac1{\sqrt2}=0.7071068

Answer

Answer

The mode is approximately 0.71.

0.71(C)\boxed{0.71\quad\text{(C)}}