Independent solution

How to solve this Pareto-Type Moments question

Setup

Setup

The density is a Pareto-type density on values above one. A finite mean requires the parameter to exceed two, and integrating the value times the density gives the displayed mean.

E[X]=(p−1)∫1∞x1−p dx,p>2E[X]=(p-1)\int_1^\infty x^{1-p}\,dx,\quad p>2

Model

Model

Equate the resulting mean expression to the stated value of two.

E[X]=p−1p−2=2E[X]=\frac{p-1}{p-2}=2

Compute

Compute

Solving the linear equation gives a parameter value of three, which satisfies the required condition for a finite mean.

p−1=2p−4,p=3p-1=2p-4,\quad p=3

Answer

Answer

The density parameter is 3.

3(C)\boxed{3\quad\text{(C)}}