Suppose we observe $X_1, X_2, ... , X_n$ i.i.d Bernoulli random variables. One then modeled how many $1s$ in those examples as a random variable, say $N_1$. Given fixed $\theta$ ($P(X_i=1)$), how one can compute $P_\theta (N_1=m)$, for $m \in \{1, 2, ..., n\}$ ?
Any hint would be appreciated.
The sum of $n$ Bernoulli random variables, each with probability $\theta$, becomes a binomial random variable with parameters $n$ and $\theta$.
We have
$$P(N_1=m)={n \choose m}\theta^m (1-\theta)^{n-m}$$