What Range of Exponents Keeps Expected Values Finite in Uniform Distributions?

[gjones89]

Homework Statement

The random variable X has uniform density on the interval [0,2], so that p(x)=1/2 for x in the interval [0,2] and p(x)=0 otherwise. Give the range of a (between minus/plus infinity) such that E[X^a] < infinity.

Homework Equations

The Attempt at a Solution

I integrated (1/2)x^a on the interval 0 to 2. This gave me an answer of (2^a)/(a+1). The only way I can see this is infinite is if a=-1. Therefore is it correct to say that E[X^a] is finite for all values of a except -1?

 

[mighty2000]

Your solution is correct. The range of a for which E[X^a] is finite is all values of a except -1. This is because when a=-1, the integral becomes ln(x) evaluated from 0 to 2, which is infinite. This means that the expected value of X^a is finite for all values of a except -1.