# Joint Distribution of sum of Bernoulli stochastic processes

Let $$X_1, X_2, \ldots$$ be independent and identically distributed (iid) Bernoulli random variables. For each $$k,$$ $$P\{X=k\} = p, \text{ } 0 Set $$N_0 = 0$$ and $$t=1,2, \ldots,$$ set $$N=\sum_{k=1}^{t} X_k.$$

Find the joint distribution of $$P\{N_{t_1} = i_1, \ldots, N_{t_d} =i_d \}$$

• Repost of math.stackexchange.com/q/3733385/321264. – StubbornAtom Jun 25 '20 at 22:04
• @StubbornAtom. That one contained errors and it was closed for that reason. The one here is the updated version – Ab2020 Jun 25 '20 at 22:26
• You are supposed to edit the previous post to add context. It will then be reopened. – StubbornAtom Jun 26 '20 at 5:59
• I'm sorry for that – Ab2020 Jun 26 '20 at 13:46

Assuming $$t_1, one can write $$P(N_{t_1}=i_1,\ldots,N_{t_d}=i_d)=P(N_{t_1}=i_1,N_{t_2}-N_{t_1}=i_2-i_1,\ldots)$$. Each RV of the form $$N_{t_k}-N_{t_{k-1}}$$ is $$Bin(t_k-t_{k-1},p)$$ independent of the others.
To conclude, the probability is $$\prod {t_k-t_{k-1} \choose i_k-i_{k-1}}p^{i_k-i_{k-1}}(1-p)^{t_k-t_{k-1}-(i_k-i_{k-1})}=p^{i_d}(1-p)^{t_d-i_d}\prod {t_k-t_{k-1} \choose i_k-i_{k-1}}$$
• Is it possible to also find $P\{ N_{t_1}-i_1, \ldots, N_{t_d} \}$? – Ab2020 Jun 25 '20 at 22:15