Give an example of a random variable X for which EX is finit

Give an example of a random variable X for which E[X] is finite, but E[X2 ] = . Make sure to give the range of the random variable, and to provide either a probability mass function or density function. Finally, make sure you verify that the random variable has the desired properties.

Solution

one of the answers can be as follows:

Pareto distribution with parameters a,x₀ which are both positive. The distribution is given by:

f_X(x) = a x₀^a / x^a+1 when x > x₀

= 0 when x < x₀

Note that E[X]=infinity, for a 1 and is finite elsewhere.

The variance is not finite for a [1,2)

Hence it satisfies your question for (1,2)

Hope this helps.