Binomial distribution

The binomial distribution is our model for the total number of successes in \(n\) independent binary trials, each with identical probability of success \(p\in[0,\,1]\).

Basic properties

Notation	\(X\sim\text{Binom}(n,\,p)\)
Range	\(\{0,\,1,\,2,\,...,\,n-1,\,n\}\)
Parameter space	\(p\in[0,\,1]\)
PMF	\(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
Expectation	\(np\)
Variance	\(np(1-p)\)

R commands

Here is the documentation for the suite of commands that let you work with the binomial distribution in R:

dbinom(x, size, prob) # PMF: P(X = x)
pbinom(q, size, prob) # CDF: P(X <= q)
qbinom(p, size, prob) # quantile function (inverse CDF)
rbinom(n, size, prob) # random numbers

Play around!

Notice that, for all values of \(p\in(0,\,1)\), the shape of the PMF becomes more like the bell curve as \(n\) grows:

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library(shiny)

discrete_pmf <- function(x, p, xlim = c(min(x) - 1, max(x) + 1), label = "", add_mean = FALSE){
  plot(x, p,
       pch = 19,
       cex = 1,
       xlab = "",
       ylab = "",
       main = "",
       ylim = c(0, 1.05 * max(p)),
       yaxs = "i",
       yaxt = "n",
       xlim = xlim,
       xaxt = "n",
       bty = "n"
  )
  segments(x,
           rep(0, length(x) + 1),
           x1 = x,
           y1 = p,
           lwd = 3
  )
  axis(1, at = floor(xlim[1]):ceiling(xlim[2]), cex.axis = 1)
  axis(2, at = seq(0, 1, length.out = 11), las = 1, cex.axis = 1.5)
  legend("topright", label, bty = "n", cex = 3)
  if(add_mean == TRUE){
    mtext("E(X)", side = 1, at = sum(x * p), col = "red", line = 2)
  }
}

discrete_cdf <- function(x, p, xlim = c(min(x) - 1, max(x) + 1), label = ""){
  closeddot = cumsum(p)
  opencircle = c(0, closeddot[1:length(x)-1])
  plot(x, closeddot, pch = 19, cex = 1,
       ylim = c(0, 1),
       ylab = "", main = "", xlab = "",
       yaxt = "n",
       xlim = xlim,
       xaxt = "n",
       #yaxs = "i", 
       bty = "n")
  points(x, opencircle, cex = 1)
  segments(c(xlim[1], x), c(0, closeddot), c(x, xlim[2]), c(0, closeddot), lwd = 1)
  axis(1, at = floor(xlim[1]):ceiling(xlim[2]), cex.axis = 1)
  axis(2, at = seq(0, 1, length.out = 11), las = 1, cex.axis = 1.5)
  legend("bottomright", label, bty = "n", cex = 3)
}

# Define UI for application that draws a histogram
ui <- fluidPage(
  
  # Application title
  titlePanel("Binomial distribution CDF and PMF"),
  
  # Sidebar with a slider input for number of bins 
  sidebarLayout(
    sidebarPanel(
      sliderInput("p",
                  "Probability of success (p):",
                  min = 0,
                  max = 1,
                  value = 0.5,
                  step = 0.01),
      sliderInput("n",
                  "Number of trials (n):",
                  min = 1,
                  max = 100,
                  value = 5,
                  step = 1)
    ),
    
    # Show a plot of the generated distribution
    mainPanel(
      plotOutput("distPlot")
    )
  )
)

# Define server logic required to draw a histogram
server <- function(input, output) {
  
  output$distPlot <- renderPlot({
    
    p <- input$p
    n <- input$n

    par(mfrow = c(2, 1), mar = c(4, 4, 2, 2))
    
    discrete_cdf(0:n, dbinom(0:n, n, p))
    discrete_pmf(0:n, dbinom(0:n, n, p), add_mean = TRUE)
    
  })
}

# Run the application 
shinyApp(ui = ui, server = server)