Bernoulli distribution

The Bernoulli distribution is one of the simplest distributions, used primarily as a “building block” to define more complicated distributions such as the Binomial. However, it’s worth treating on its own.

Assumptions

Let \(X\) be a discrete random variable that can only take on values of 0 or 1. \(X\) will be a Bernoulli variable with parameter \(p\) if and only if,

  • \(P(X=1) = p\)
  • \(P(X=0) = 1 - p\)1

That’s it! A Bernoulli variable is a simple 1/0 switch with fixed probability of being 1. A coin flip is a Bernoulli variable, including flips of a weighted coin. Any single attempt with two outcomes (success or failure, life or death, profit or loss) is a Bernoulli variable.

Definition

\[\begin{array}{ll} \text{Support:} & \{0,1\} \\ \text{Parameter(s):} & p,\text{ the probability of success }(p \in [0,1]) \\ \text{PMF:} & P(X=k) = \left\{\begin{array}{cl} 1-p, & \quad k=0 \\ p, & \quad k=1 \end{array}\right\} \\ \text{CDF:} & F_X(x) = \left\{\begin{array}{cl} 0, & \quad x \lt 0 \\ 1-p, & \quad 0 \le x \lt 1 \\ 1, & \quad x \ge 1 \end{array}\right\} \\ \text{Mean:} & \mathbb{E}[X]=p \\ \text{Variance:} & \mathbb{V}[X]=p(1-p) \\ \end{array}\]

Visualizer

#| '!! shinylive warning !!': |
#|   shinylive does not work in self-contained HTML documents.
#|   Please set `embed-resources: false` in your metadata.
#| standalone: true
#| viewerHeight: 650

library(shiny)
library(bslib)

ui <- page_fluid(
      tags$head(tags$style(HTML("body {overflow-x: hidden;}"))),
  title = "Bernoulli distribution PMF",
  fluidRow(plotOutput("distPlot")),
  fluidRow(sliderInput("p", "Probability (p)", min=0, max=1, step=0.01, value=0)))

server <- function(input, output) {
  output$distPlot <- renderPlot({
    plot(x=c(0,1),y=c(1-input$p,input$p),main=NULL,
         xlim=c(0,1),ylim=c(0,1),
         xlab='x',ylab='Probability',type='h',lwd=3)})
}

shinyApp(ui = ui, server = server)

Relations to other distributions

  • The sum of \(n\) identically and independently-distributed Binomial variates is distributed Binomial(\(n,p\)).

    • As a corollary, for \(n=1\), a Binomial variable is simply a Bernoulli variable.

  • The number of identical and independently-distributed Bernoulli trials required until the first success (\(X=1\)) is a Geometric variable with the same parameter \(p\).

  • The number of identical and independently-distributed Bernoulli trials required until the \(r\)th success is Negative Binomial-distributed with parameters \(r\) and \(p\).

  1. Sometimes statisticians use the convention \(q = 1 - p\).↩︎