Computing mean and variance of custom distribution

Question

How can we compute the mean and variance of some custom distribution such as:

$P(X=a) = \frac{1}{a(a+1)}$ for $a=1,2,3,...$

What I have tried so far:

Mean: $ \mu = \sum_{a=1}^\infty a \cdot \frac{1}{a(a+1)} $

This leads to : $ \mu = \sum_{a=1}^\infty \frac{1}{(a+1)} $

So, does this series converge?

Something similar should follow for the variance, but then, I will need the mean.

Any ideas?

Thanks..!

Answer 1

You are right about the expression for the mean, but $ \sum_{a=1}^\infty \frac{1}{a+1}$ diverges by comparison with the harmonic series. So this distribution has an undefined mean. And distributions that have an undefined mean have an undefined variance.