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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.

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  • $\begingroup$ 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. $\endgroup$ – Remy Mar 12 '18 at 22:15
  • $\begingroup$ yes, you're right! I edit it :) $\endgroup$ – user540113 Mar 12 '18 at 22:15
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The sum of $n$ Bernoulli random variables, each with probability $\theta$, becomes a binomial random variable with parameters $n$ and $\theta$.

We have

$$P(N_1=m)={n \choose m}\theta^m (1-\theta)^{n-m}$$

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