Factsheet: Exponential distribution

Statistics

Author

Michelle Arnetta and Tom Coleman

Summary

A factsheet on the exponential distribution.

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

ui <- page_fluid(
  title = "Exponential distribution calculator",
  
  layout_columns(
    col_widths = c(4, 8),
    
    # Left column - Inputs
    card(
      card_header("Parameters"),
      card_body(
        numericInput("rate", "Rate parameter (λ):", value = 0.5, min = 0.01, step = 0.01),
        # Removed the helpText about mean and variance
        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 = 10, value = 2, step = 0.1)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'greater'",
          sliderInput("x_greater", "x value:", min = 0, max = 10, value = 2, step = 0.1)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'between'",
          sliderInput("x_lower", "Lower bound (x):", min = 0, max = 10, value = 1, step = 0.1),
          sliderInput("x_upper", "Upper bound (y):", min = 0, max = 10, value = 3, step = 0.1)
        )
      )
    ),
    
    # Right column - Plot
    card(
      card_header("Exponential 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) {
  
  # When rate changes, adjust the range of sliders
  observe({
    # For exponential distribution with parameter rate, the mean is 1/rate
    # Set a reasonable max value based on this
    max_x <- max(round(5 / input$rate, 1), 10)
    
    updateSliderInput(session, "x_less", max = max_x)
    updateSliderInput(session, "x_greater", max = max_x)
    updateSliderInput(session, "x_lower", max = max_x)
    updateSliderInput(session, "x_upper", max = max_x)
  })
  
  # 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$plot_title <- renderUI({
    title <- sprintf("Exponential(λ = %.2f)", input$rate)
    tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
  })
  
  # Calculate the probability based on user selection
  probability <- reactive({
    if (input$prob_type == "less") {
      prob <- pexp(input$x_less, rate = input$rate)
      explanation <- sprintf("P(X ≤ %.1f) = %.6f or %.4f%%", 
                           input$x_less, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "less", x = input$x_less))
      
    } else if (input$prob_type == "greater") {
      prob <- 1 - pexp(input$x_greater, rate = input$rate)
      explanation <- sprintf("P(X ≥ %.1f) = %.6f or %.4f%%", 
                           input$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$x_upper) {
        # For continuous distributions, P(X = a) = 0
        prob <- 0
      } else {
        upper_prob <- pexp(input$x_upper, rate = input$rate)
        lower_prob <- pexp(input$x_lower, rate = input$rate)
        prob <- upper_prob - lower_prob
      }
      explanation <- sprintf("P(%.1f ≤ X ≤ %.1f) = %.6f or %.4f%%", 
                           input$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 Exponential distribution plot
  output$distPlot <- renderPlot({
    # Determine the range for the x-axis
    rate <- input$rate
    max_x <- max(round(5 / rate, 1), 10)
    
    # Create data frame for plotting
    x_values <- seq(0, max_x, length.out = 500)
    density_values <- dexp(x_values, rate = rate)
    df <- data.frame(x = x_values, density = density_values)
    
    # Create base plot
    p <- ggplot(df, aes(x = x, y = density)) +
      geom_line(size = 1, color = "darkgray") +
      labs(x = "Time (X)", y = "probability density function") +
      theme_minimal() +
      theme(panel.grid.minor = element_blank())
    
    # 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 <- dexp(fill_x, rate = rate)
      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, max_x, length.out = 200)
      fill_y <- dexp(fill_x, rate = rate)
      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 <- dexp(fill_x, rate = rate)
      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 exponential distribution is used when \(X\) is the waiting time before a certain event occurs. It is similar to the geometric distribution, but the exponential distribution uses a continuous waiting time instead of the integer number of trials.

Notation: \(X \sim \textrm{Exponential}(\lambda)\) or \(X \sim \textrm{Exp}(\lambda)\)

Parameter: An integer \(\lambda\), representing number of times an event occurs within a specific period of time.

Quantity	Value	Notes
Mean	\(\mathbb{E}(X) = \frac{1}{\lambda}\)
Variance	\(\mathbb{V}(X) = \frac{1}{\lambda^2}\)
PDF	\(\mathbb{P}(X=x)=\lambda e^{-\lambda x}\)
CDF	\(\mathbb{P}(X \leq x)=1-e^{-\lambda x}\)

Example: Customers enter Cantor’s Confectionery at an average rate of 20 people per hour, and the time distance between each visit can be modelled by an exponential distribution. This can be expressed as \(X \sim \textrm{Exp}(20)\).

Factsheet: Exponential distribution

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