When we have $$m$$ i.i.d. samples $$\left\{x^{(1)}, \dots, x^{(m)}\right\}$$ from a Bernoulli distribution with parameter $$\theta$$, such that $$P(x^{(i)},\theta) = \theta^{x^{(i)}} (1-\theta)^{(1-x^{(i)})}$$, an 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 $$\theta$$

For a Bernoulli distribution characterizing a random variable $$ X $$ with probability $$ p $$ for $$ X=1 $$ ("success") and $$ 1-p $$ for $$ X=0 $$ ("failure"), the key characteristics are:
1) Probability mass function: $$ P(X) = p^{X}(1-p)^{1-X} $$
2) Expectation: $$ E[X] = p(1) + (1-p)(0) = p $$
3) Variance: $$ Var(X) = E[X^2] - (E[X])^2 = p - p^2 = p(1-p) $$

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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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