Bernoulli Distribution

This is a Discrete Distribution. This is about finding the probability of a single trial outcome.

In bernoulli distribution, the trial outcome must be boolean. (yes/no, 1/0, red/green)

1. The Core Idea

If an experiment has only two outcomes:

• Success ($1$) happens with probability $p$
• Failure ($0$) happens with probability $q = 1 - p$

A random variable $X$ that follows a Bernoulli distribution is written as:

$$X \sim \text{Bernoulli}(p)$$

Classic Examples:

• Flipping a fair coin once: $p = 0.5$ (Heads = 1, Tails = 0)
• Converting a website visitor into a sale: $p = 0.03$ (Sale = 1, No sale = 0)
• Passing a test: $p = 0.85$ (Pass = 1, Fail = 0)

PMF

$$P(X = k) = p^k (1 - p)^{1 - k}$$

• $k$ is boolean value. so either ($1$ or $0$)
• $p$ is the probability of $k$ happen

Properties

Mean (Expected Value)

$$E[X] = p$$

Variance

$$\text{Var}(X) = p(1 - p)$$

Importance

• This distribution mainly act as an atomic block for many other distributions.
• comes in handy in Logistic Regression