Interactive: Mean and variance of beta distribution

Summary

Interactive diagram for seeing how the mean and variance affect the shape of the beta distribution.

#| '!! shinylive warning !!': |
#|   shinylive does not work in self-contained HTML documents.
#|   Please set `embed-resources: false` in your metadata.
#| standalone: true
#| viewerHeight: 770

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

ui <- page_fluid(
  title = "Beta distribution",
  
  # Plot at the top
  card(
    card_header("Beta distribution plot"),
    card_body(
      uiOutput("plot_title"),
      plotOutput("distPlot", height = "400px")
    )
  ),
  
  # Parameters below
  card(
    card_header("Distribution parameters"),
    card_body(
      layout_columns(
        col_widths = c(6, 6),
        sliderInput("mean", "Mean:", min = 0.1, max = 0.9, value = 0.5, step = 0.01),
        sliderInput("variance", "Variance:", min = 0.01, max = 0.2, value = 0.05, step = 0.01)
      )
    )
  )
)

server <- function(input, output, session) {
  
  # Convert mean and variance to shape parameters
  beta_params <- reactive({
    mean_val <- input$mean
    var_val <- input$variance
    
    # For a beta distribution: mean = α/(α+β), var = αβ/((α+β)²(α+β+1))
    # Solving for α and β given mean and variance:
    # α = mean * ((mean * (1 - mean)) / variance - 1)
    # β = (1 - mean) * ((mean * (1 - mean)) / variance - 1)
    
    # Ensure variance doesn't exceed theoretical maximum
    max_var <- mean_val * (1 - mean_val) / 1.01  # Small buffer
    if (var_val >= max_var) {
      var_val <- max_var
    }
    
    common_term <- (mean_val * (1 - mean_val)) / var_val - 1
    alpha <- mean_val * common_term
    beta <- (1 - mean_val) * common_term
    
    # Ensure positive parameters
    alpha <- max(alpha, 0.01)
    beta <- max(beta, 0.01)
    
    return(list(alpha = alpha, beta = beta))
  })
  
  # Display the plot title with distribution parameters
  output$plot_title <- renderUI({
    params <- beta_params()
    title <- sprintf("Beta(α = %.2f, β = %.2f)", params$alpha, params$beta)
    subtitle <- sprintf("Mean = %.2f, Variance = %.4f", input$mean, input$variance)
    div(
      tags$h4(title, style = "text-align: center; margin-bottom: 5px;"),
      tags$p(subtitle, style = "text-align: center; color: gray; margin-bottom: 15px;")
    )
  })
  
  # Generate the beta distribution plot
  output$distPlot <- renderPlot({
    # Get shape parameters
    params <- beta_params()
    shape1 <- params$alpha
    shape2 <- params$beta
    
    # Create data frame for plotting with fixed x range
    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 plot with fixed axes
    p <- ggplot(plot_df, aes(x = x, y = density)) +
      geom_line(size = 1, color = "#3F6BB6") +
      geom_area(aes(x = x, y = density), fill = "#3F6BB6", alpha = 0.3) +
      labs(x = "X", y = "Probability Density") +
      theme_minimal() +
      theme(panel.grid.minor = element_blank()) +
      xlim(0, 1) +
      ylim(0, 10) +  # Fixed y-axis range
      # Add reference line at mean
      geom_vline(xintercept = input$mean, linetype = "dashed", color = "#DB4315", alpha = 0.7) +
      annotate("text", x = input$mean + 0.08, y = 8.5, label = "Mean", color = "#DB4315", size = 4)
    
    return(p)
  })
}

shinyApp(ui = ui, server = server)

Further reading

This interactive element appears in Guide: Expected value, variance, standard deviation. Please click this link to go to the guide.

Version history

v1.0: initial version created 08/24 by tdhc.

This work is licensed under CC BY-NC-SA 4.0.