Binomial Parameters

The binomial distribution can be thought of as a number of draws, \(n\), from an urn with a proportion \(p\), of black balls.

The probability of drawing exactly \(x\) balls from an this urn is: \[ p(X|n,p) = \binom{n}{X} p^X (1-p)^{n-X}\]

The expected value is \(np\), and the standard deviation is \(\sqrt{np(1-p)}\).

Sometimes we write this in terms of the proportion of black balls in the sample. That is \(p\), with a standard deviation of \(\sqrt{p(1-p)/n}\).

#| standalone: true
#| viewerHeight: 600
library(shiny)
library(ggplot2)
ui <- fluidPage(
inputPanel(
  sliderInput("n", label = "Number of draws:",
              min=0, max=100, value=10, step=1),
  
  sliderInput("p", label = "Probability of success:",
              min = 0, max = 1, value = .6, step = 0.01)
),
mainPanel(
  plotOutput("bincurve")))

server <- function (input,output) {
  output$bincurve <- renderPlot({
  n <- as.numeric(input$n)
  p <- as.numeric(input$p)
  dat <- data.frame(x=0:n,y=dbinom(0:n,n,p))
  ggplot(dat,aes(x,y)) +geom_col()  

})
}
shinyApp(ui=ui,server=server)

Note that this distribution is positively skewed if \(p < 0.5\) and negatively skewed if \(p > 0.5\).

Note how when \(n\) gets large, the binomial distribution looks a lot like the normal. This is one of the first central limit theorems that was discovered. (The closer that \(p\) is to 0 or 1, the longer convergence to the normal takes.)