> [!definition] > > The Bernoulli distribution $B(1, p)$ is a discrete [[Random Variable|random variable]] with range $R = \{0, 1\}$ that represents the number of successes with [[Probability|probability]] $p$ after a single [[Bernoulli Process|Bernoulli trial]]. It has the following [[Probability Mass Function|probability mass function]]: > $ > p(1) = 1, p(0) = 1 - p = q \quad \text{where } 0 \le p \le 1 > $ > [!theorem] [[Entropy]] > > $ > H_b(p) = - p \log_2(p) - q\log_2(q) > $ > $H_b(p)$ is greatest when $p = \frac{1}{2}$, meaning that the Bernoulli variable has the greatest uncertainty when it is symmetrical. > > It is a special case of the [[Binomial Random Variable|binomial distribution]]. A Bernoulli distribution is [[Uniform Distribution|uniform]] if and only if $p = \frac{1}{2}$, in which case it is known as the symmetric Bernoulli distribution.