Question

Let N have a Poisson distribution with parameter lander=1. Conditioned on N=n,
let X have a uniform distribution over the integers 0,1,.......,n+1. What is the
marginal distribution for X?

Step by step and show what definition you use

Answer 1

We are given the distribution of N here as:

Given N = n, the distribution of X here is given as:

The marginal distribution of X here is obtained as:

Similarly for other values of X also we obtain the same probabilities. The general marginal PDF here is obtained as:

This is the required marginal PDF for X here.