a Is there a discrete distibution with support 123 such that

a. Is there a discrete distibution with support 1,2,3..., such that the value of the PMF at n is proportional to 1/n?

b. Is there a discrete disribution with support 1,2,3..., such that the value of the PMF at n is proportional to 1/(n^2)?

Solution

a)For it to be valid distribution, we need sum of all probabilities equal to 1. Since sum(1/n) expands, it cannot be equal to a constant value. Had this been a constant, the PMF would have been 1/constant * 1/n., whose some is 1. But since the sum if not constant (it expands), this PMF is not possible

b) Sum(1/n^2) does infact converge to pi^2/6, so we can create random variable with PMF:

P(X=n) = 6/pi^2 * 1/n^2 for n =1,2,3..., and zero otherwise.

Note that the sum is 1, P(X) is non-negative, so this is a valid probability distribution.