Let $X_1, X_2, \ldots$ be independent and identically distributed (iid) Bernoulli random variables. For each $k,$ $$P\{X=k\} = p, \text{ } 0 <p<1.$$ Set $N_0 = 0$ and $t=1,2, \ldots,$ set $$N=\sum_{k=1}^{t} X_k.$$
Find the joint distribution of $$P\{N_{t_1} = i_1, \ldots, N_{t_d} =i_d \}$$
Assuming $t_1<t_2<\ldots<t_d$, 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 $\prod {t_k-t_{k-1} \choose i_k-i_{k-1}}p^{i_k-i_{k-1}}(1-p)^{t_k-t_{k-1}-(i_k-i_{k-1})}=p^{i_d}(1-p)^{t_d-i_d}\prod {t_k-t_{k-1} \choose i_k-i_{k-1}}$