Bernoulli distribution
High school
Consider a random experiment that either succeeds or fails, such as a coin flip that succeeds if the coin lands heads. A random variable $X$ that equals $1$ for success and $0$ for failure has a Bernoulli distribution with success probability $p$ if
\[P(X = 1) = p \quad \text{and} \quad P(X = 0) = 1 - p.\]Examples
Worked examples
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Customers arriving at a shop independently of each other at an average rate of $3$ per hour
: Whether at least one customer arrives in an hour: Bernoulli with $p = 1 - e^{-3} \approx 0.950$
Undergraduate
For customers arriving at an average rate of $3$ per hour, the indicator that at least one customer arrives in an hour has a Bernoulli distribution with success probability $1 - P(\text{no customers}) = 1 - e^{-3}$.
Properties
- A Bernoulli random variable with success probability $p$ has mean $p$ and variance $p (1 - p)$.