Factsheet: Beta distribution

Statistics

Author

Michelle Arnetta and Tom Coleman

Summary

A factsheet about the beta distribution.

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library(shiny)
library(bslib)
library(ggplot2)

ui <- page_fluid(
  title = "Beta distribution calculator",
  
  layout_columns(
    col_widths = c(4, 8),
    
    # Left column - Inputs
    card(
      card_header("Parameters"),
      card_body(
        numericInput("shape1", "Shape parameter α:", value = 2, min = 0.01, step = 0.1),
        numericInput("shape2", "Shape parameter β:", value = 2, min = 0.01, step = 0.1),
        hr(),
        radioButtons("prob_type", "Probability to calculate:",
                    choices = list("P(X ≤ x)" = "less", 
                                  "P(X ≥ x)" = "greater", 
                                  "P(x ≤ X ≤ y)" = "between"),
                    selected = "less"),
        conditionalPanel(
          condition = "input.prob_type == 'less'",
          sliderInput("x_less", "x value:", min = 0, max = 1, value = 0.5, step = 0.01)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'greater'",
          sliderInput("x_greater", "x value:", min = 0, max = 1, value = 0.5, step = 0.01)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'between'",
          sliderInput("x_lower", "Lower bound (x):", min = 0, max = 1, value = 0.25, step = 0.01),
          sliderInput("x_upper", "Upper bound (y):", min = 0, max = 1, value = 0.75, step = 0.01)
        )
      )
    ),
    
    # Right column - Plot
    card(
      card_header("Beta distribution plot"),
      card_body(
        uiOutput("plot_title"),
        plotOutput("distPlot", height = "300px")
      )
    )
  ),
  
  # Bottom row - Results
  card(
    card_header("Results"),
    card_body(
      textOutput("explanation")
    )
  )
)

server <- function(input, output, session) {
  
  # Ensure that x_upper is always greater than or equal to x_lower
  observe({
    if (input$x_upper < input$x_lower) {
      updateSliderInput(session, "x_upper", value = input$x_lower)
    }
  })
  
  # Display the plot title with distribution parameters
  output$plot_title <- renderUI({
    title <- sprintf("Beta(α = %.2f, β = %.2f)", input$shape1, input$shape2)
    tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
  })
  
  # Calculate the probability based on user selection
  probability <- reactive({
    if (input$prob_type == "less") {
      prob <- pbeta(input$x_less, shape1 = input$shape1, shape2 = input$shape2)
      explanation <- sprintf("P(X ≤ %.2f) = %.6f or %.4f%%", 
                           input$x_less, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "less", x = input$x_less))
      
    } else if (input$prob_type == "greater") {
      prob <- 1 - pbeta(input$x_greater, shape1 = input$shape1, shape2 = input$shape2)
      explanation <- sprintf("P(X ≥ %.2f) = %.6f or %.4f%%", 
                           input$x_greater, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "greater", x = input$x_greater))
      
    } else if (input$prob_type == "between") {
      if (input$x_lower == input$x_upper) {
        # For continuous distributions, P(X = a) = 0
        prob <- 0
      } else {
        upper_prob <- pbeta(input$x_upper, shape1 = input$shape1, shape2 = input$shape2)
        lower_prob <- pbeta(input$x_lower, shape1 = input$shape1, shape2 = input$shape2)
        prob <- upper_prob - lower_prob
      }
      explanation <- sprintf("P(%.2f ≤ X ≤ %.2f) = %.6f or %.4f%%", 
                           input$x_lower, input$x_upper, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "between", 
                 lower = input$x_lower, upper = input$x_upper))
    }
  })
  
  # Display an explanation of the calculation
  output$explanation <- renderText({
    res <- probability()
    return(res$explanation)
  })
  
  # Generate the beta distribution plot
  output$distPlot <- renderPlot({
    # Get parameters
    shape1 <- input$shape1
    shape2 <- input$shape2
    
    # Create data frame for plotting
    # Beta distribution is defined on the interval [0, 1]
    x_values <- seq(0, 1, length.out = 500)
    density_values <- dbeta(x_values, shape1 = shape1, shape2 = shape2)
    plot_df <- data.frame(x = x_values, density = density_values)
    
    # Create base plot
    p <- ggplot(plot_df, aes(x = x, y = density)) +
      geom_line(size = 1, color = "darkgray") +
      labs(x = "X", y = "probability density function") +
      theme_minimal() +
      theme(panel.grid.minor = element_blank()) +
      xlim(0, 1) +
      # Adjust y-limit based on maximum density to handle tall peaks
      ylim(0, max(density_values) * 1.05)
    
    # Add shaded area based on selected probability type
    res <- probability()
    
    if (res$type == "less") {
      # Create data for the filled area
      fill_x <- seq(0, res$x, length.out = 200)
      fill_y <- dbeta(fill_x, shape1 = shape1, shape2 = shape2)
      fill_df <- data.frame(x = fill_x, density = fill_y)
      
      p <- p + geom_area(data = fill_df, aes(x = x, y = density), 
                        fill = "#3F6BB6", alpha = 0.6)
      
    } else if (res$type == "greater") {
      # Create data for the filled area
      fill_x <- seq(res$x, 1, length.out = 200)
      fill_y <- dbeta(fill_x, shape1 = shape1, shape2 = shape2)
      fill_df <- data.frame(x = fill_x, density = fill_y)
      
      p <- p + geom_area(data = fill_df, aes(x = x, y = density), 
                        fill = "#3F6BB6", alpha = 0.6)
      
    } else if (res$type == "between") {
      # Create data for the filled area
      fill_x <- seq(res$lower, res$upper, length.out = 200)
      fill_y <- dbeta(fill_x, shape1 = shape1, shape2 = shape2)
      fill_df <- data.frame(x = fill_x, density = fill_y)
      
      p <- p + geom_area(data = fill_df, aes(x = x, y = density), 
                        fill = "#3F6BB6", alpha = 0.6)
    }
    
    return(p)
  })
}

shinyApp(ui = ui, server = server)

Where to use: The beta distribution is used to model the distribution of probabilities or proportions. Hence, the random variable \(0 \leq X \leq 1\).

Notation: \(X \sim \textrm{Beta}(\alpha,\beta)\)

Parameters: Two positive real numbers \(\alpha,\beta\), which are shape parameters. These can be specified as follows in terms of \(n\) and \(k\) where \(n\) is the number of Bernoulli trials and \(k\) is the number of successes:

\(\alpha = k + 1\)
\(\beta = n - k + 1\)

Quantity	Value	Notes
Mean	\(\mathbb{E}(X) = \dfrac{\alpha}{\alpha+\beta}\)
Variance	\(\mathbb{V}(X) = \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)
PDF	\(\mathbb{P}(X=x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{\textrm{B}(\alpha,\beta)}\)	\(\textrm{B}(x,y)\) is the beta function
CDF	\(\mathbb{P}(X \leq x)=I_{x}(\alpha,\beta)\)	\(I_{x}(a,b)\) is the regularized incomplete beta function

Example: Cantor’s Confectionery is visited by 10 customers, and 6 of them purchase something from the store. Taking the buying customers as successes and the total visiting customers as number of trials, there would be 6 successes, allowing you to find the following parameters:

\(\alpha = 6 + 1 = 7\)
\(\beta = 10 - 6 + 1 = 5\)

Then the distribution of the probabilities of a customer purchasing from Cantor’s Confectionery can be expressed as \(X \sim \textrm{Beta}(7,5)\), meaning the first shape parameter is 7 and the second shape parameter is 5.

Factsheet: Beta distribution

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