Independent solution

How to solve this Continuous Distributions question

Setup

Setup

Write the density as k times the inverse fourth power of 1+x on the positive half-line. The constant k must be determined before its first moment can be evaluated.

fX(x)=k(1+x)−4,x>0f_X(x)=k(1+x)^{-4},\qquad x>0

Model

Model

First impose total mass one to obtain k. Then insert that constant into the defining first-moment integral.

1=k∫0∞(1+x)−4 dx=k31=k\int_0^{\infty}(1+x)^{-4}\,dx=\frac{k}{3}
E[X]=3∫0∞x(1+x)−4 dxE[X]=3\int_0^{\infty}x(1+x)^{-4}\,dx

Compute

Compute

Normalization gives k=3. The remaining beta-type integral is 1/2-1/3=1/6, and multiplication by 3 gives E[X]=1/2.

k=3k=3
E[X]=3(12−13)=12E[X]=3\left(\frac12-\frac13\right)=\frac12

Answer

Answer

Hence the mean of X is 1/2, which corresponds to choice C.

12(C)\boxed{\frac12\quad\text{(C)}}