Find the expectation of X^p for every p that is a real number

[sillybuffalo]

The random variable X has the probability density function given by:


for x > 1

and


for x < or equal to 1.

The question asks me to find the expectation of the power X^p, for every p that is a real number.

I know that the expectation of the function of a random variable is given by:



where g(x) = x^p, and f_x(x) = c/(x^4)

Using the limits of integration from 1 to infinity, I was able to find

E[X^p] = -3 / (p-3) for p < 3.

However, I am having difficulty solving the integral for p = 3 and p >3, because the integral does not converge.

What am I doing wrong here?

 

[DrPhil]

The random variable X has the probability density function given by:

View attachment 3417
for x > 1

and

View attachment 3418
for x < or equal to 1.

The question asks me to find the expectation of the power X^p, for every p that is a real number.

I know that the expectation of the function of a random variable is given by:

View attachment 3416

where g(x) = x^p, and f_x(x) = c/(x^4)

Using the limits of integration from 1 to infinity, I was able to find

E[X^p] = -3 / (p-3) for p < 3.

However, I am having difficulty solving the integral for p = 3 and p >3, because the integral does not converge.

What am I doing wrong here?

The only thing you are doing wrong is not to trust your results. It should not surprise you that the integral does not converge if p is too great.

\(\displaystyle \displaystyle E[x^p] = \begin{cases}\dfrac{3}{3-p} & p<3 \\ \text{not defined} & p \ge 3 \end{cases} \)