Let $n \in \Bbb N$ and the probability spaces $(\Omega_k, \mathcal {F_k}, \Bbb P_{k}^{p})$, be given by $\Omega_k := {0,1}$, $\mathcal {F_k} := 2^{\Omega_k}$ and $\Bbb P_{k}^{p} := p\delta_1 + (1-p)\delta_0$. Let $\Omega := \prod_{k\in \Bbb N}\Omega_k$, $\mathcal F := ⊗_{k\in \Bbb N}\mathcal {F_k}$ and $P^{p}:=⊗_{k\in \Bbb N}P_{k}^{p}$.
- I need to determine the distribution of the following r.v. $X_n, n\in \Bbb N$, on $(\Omega, \mathcal {F}, \Bbb P^{p})$ $$X_n: \Omega \to \{0,...n\}: \omega \mapsto |i\in \{1,...n\}|\omega_i = 1|$$
- Here it goes other way around. Given, that a r.v. on $(\Omega, \mathcal {F}, \Bbb P^{p})$ is Geom($p$)-distributed, it has to be constructed.
It seems to be obvious that in the first part we have $X_n\sim Bin(n,p)$ but I have no clue so far how to show it rigorously. Without using mathematical terminology, there are ${n \choose k}$ choices for {1}'s and {0}'s to appear k and (n-k) times respectively. Probably, if it's done, part 2 can be solved in a similar way.
Any hint would be appreciated :)