# Probability of 1s random variable on Bernoulli model

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.

• You have $n$ iid Bernoulli's which becomes a binomial. I don't think $n$ is what variable you should use in $P_{\theta}(N_1=n)$ unless you seek $n$ successes in $n$ trials.
– Remy
Mar 12, 2018 at 22:15
• yes, you're right! I edit it :)
– user540113
Mar 12, 2018 at 22:15

The sum of $n$ Bernoulli random variables, each with probability $\theta$, becomes a binomial random variable with parameters $n$ and $\theta$.
$$P(N_1=m)={n \choose m}\theta^m (1-\theta)^{n-m}$$