# Calculation of multivariate probability mass function

How to calculate the following multivariate probability mass function:

$$P(X_1-X = n, X_2-X = n, ..., X_{N-1}-X = n)$$

Where $$n$$ and $$N$$ are positive integers, and $$X_i$$ and $$X$$ are iid random variables with the following discrete probability distribution:

$$P(i)=\frac{C_{i-1}}{2^{2i-1}}$$; $$C_{i}$$ are Catalan numbers.

Looking at: Integrating pdf times cdf squared and walking backwards, from multivariate to integral representation I got:

$$\sum _{i=0} ^{+ \infty} {P(i)P(i+n)^{N-1}}$$

However, it appears that the last formula gives me correct results only for N=2. Am I doing something wrong?

• What have you tried so far? If these are independent random variables, you can work out the answer by conditioning on the value of $X$. Sep 25, 2023 at 13:05
• They are iids, I have amended the question. I have also added something I tried but didn't work. Sep 25, 2023 at 15:50

The start of your solution looks correct to me. We have \begin{align} p & = P(X_1-X = n, \ldots , X_{N-1}-X = n ) \\ & = \sum_{i=1}^\infty P(X_1-X = n, \ldots , X_{N-1}-X = n | X = i) P(X = i) \\ & = \sum_{i=1}^\infty P(X_1 = n+i, \ldots , X_{N-1} = n+i) P(X = i) \\ & = \sum_{i=1}^\infty f_X(n+i)^{N-1} f_X(i) \end{align} by independence, where $$f_X$$ is the PMF. Substituting in $$f_X(i)$$ gives $$p = \sum_{i=1}^\infty \left[\frac{C_{n+i-1}}{2^{2(n+i)-1}} \right]^{N-1} \frac{C_{i-1}}{2^{2i-1}}$$