When we have $m$ $i.i.d$ samples  $\{x^{(1)},...,x^{(m)}\}$ from a Bernoulli distribution with parameter $\theta$, such that: 
	$$P(x^{(i)},\theta) = \theta^{x^{(i)}} (1-\theta)^{(1-x^{i})}$$,
a unbiased estimator for $\theta$ is the mean of those samples:
	$\hat{\theta}_m = \frac{1}{m} \sum_{i=1}^m x^{(i)}$


Unbiased Estimator for Bernoulli Parameter

When we talk about a distribution, there are some things we might be interested in:
We assume this Bernoulli Distribution has a probabilty p, for n=1 ("success") and 1-p for n=0("failure")
N is the number of trails
1) Probability density function: $ P(N)=p^{N} *(1-p)^{1-N} $
2) Expectation: $ E(X)=p*1+(1-p)*0=p $
3) Variance:  $α=E[X^{2}]-E[X]^{2}=p-p^{2}$


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Bernoulli distribution is a distribution over a single binary variable. It can only have results in 0 or 1. The sum of the probability of 0  and the probability of 1 is equal to 1. If we denote the probability of 0 as $\sigma$, Bernoulli distribution then has a mean of $\sigma$ and a variance of  $\sigma * (1- \sigma)$.

Bernoulli Distribution

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A “Bernoulli trial” is an experiment or case where the outcome follows a Bernoulli distribution. Some common examples of Bernoulli trials include:

1) A single flip of a coin that may have a head(0) or a tail(1) outcome.

2) A single birth of either a boy(0) or a girl(1).

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