Bernoulli Distribution

A Bernoulli distribution is a distribution with two possible outcomes: the event happens or does not happen. A random variable \(\mathbf{X}\) in a Bernoulli distribution can take two values:

\[ \mathbf{X} = \{0, 1\} \]

where 0 means the event does not happen, and 1 means the event does happen.

Thus,

\[ P(\mathbf{X} = 1) = p \\ P(\mathbf{X} = 0) = q = 1 - p \]

where \(p\) indicates the probability of success (or occurrence of the event), and \(1 - p\) is the probability of failure (or non-occurrence).

Bernoulli \(\rightarrow\) Binomial

When we conduct \(n\) independent Bernoulli trials \((B_1, B_2, B_3, \dots, B_n)\), each with the same probability of success \(p\), the distribution of the sum of successes follows a Binomial Distribution.

Let \(\mathbf{X} =\) total number of successes \(= B_1 + B_2 + \dots + B_n\), where \(B_i \sim \text{Bernoulli}(p)\).

Then,

\[ \mathbf{X} \sim \text{Binomial}(n, p) \\ P(\mathbf{X} = k) = \binom{n}{k} p^k q^{n - k} \]

where \(n\) and \(p\) are the parameters representing the number of trials and the probability of success, respectively.