Proof: Probability mass function of the Bernoulli distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Bernoulli distribution ▷ Probability mass function

Theorem: Let $X$ be a random variable following a Bernoulli distribution:

$$\label{eq:Bern} X \sim \mathrm{Bern}(p) \; .$$

Then, the probability mass function of $X$ is

$$\label{eq:Bern-pmf} f_X(x) = \left\{ \begin{array}{rl} p \; , & \text{if} \; x = 1 \\ 1-p \; , & \text{if} \; x = 0 \; . \\ \end{array} \right. \; .$$

Proof: This follows directly from the definition of the Bernoulli distribution.

∎

Sources:

• original work

Metadata: ID: P96 | shortcut: bern-pmf | author: JoramSoch | date: 2020-05-11, 22:10.