Explain why it is not possible to define a uniform distribut

Explain why it is not possible to define a uniform distribution function on a countably infinite sample space

Solution

For a distribution to be defined we need

1.) To define a non negative probability for each value in the set.

2.) Have those values sum to 1

If your data set is numbers from 1-1000 we can do that. But if our data is unbounded the probability will not be defined. However small and finite the probability be the distribution will not be valid. For example if we take our probability as

e-10 and if we sum this to infinity the probabilities will be >1 which is not a valid distribution