Factsheet: Continuous uniform distribution

Statistics

Author

Michelle Arnetta and Tom Coleman

Summary

A factsheet for the continuous uniform distribution.

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

ui <- page_fluid(
  title = "Continuous uniform distribution calculator",
  
  layout_columns(
    col_widths = c(4, 8),
    
    # Left column - Inputs
    card(
      card_header("Parameters"),
      card_body(
        numericInput("a", "Minimum value (a):", value = 0, step = 0.1),
        numericInput("b", "Maximum value (b):", value = 10, 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' || input.prob_type == 'greater'",
          sliderInput("x_value", "x value:", min = 0, max = 10, value = 5, step = 0.1)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'between'",
          sliderInput("x_lower", "Lower bound (x):", min = 0, max = 10, value = 3, step = 0.1),
          sliderInput("x_upper", "Upper bound (y):", min = 0, max = 10, value = 7, step = 0.1)
        )
      )
    ),
    
    # Right column - Plot
    card(
      card_header("Continuous uniform 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 b is always greater than a
  observe({
    if (input$b <= input$a) {
      updateNumericInput(session, "b", value = input$a + 1)
    }
  })
  
  # Update the range of the sliders when a or b changes
  observe({
    updateSliderInput(session, "x_value", min = input$a, max = input$b, value = min(max(input$a, (input$a + input$b)/2), input$b))
    updateSliderInput(session, "x_lower", min = input$a, max = input$b, value = min(max(input$a, input$a + (input$b - input$a)/3), input$b))
    updateSliderInput(session, "x_upper", min = input$a, max = input$b, value = min(max(input$a, input$a + 2*(input$b - input$a)/3), input$b))
  })
  
  # 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$plot_title <- renderUI({
    title <- sprintf("Unif(a = %.1f, b = %.1f)", input$a, input$b)
    tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
  })
  
  # Calculate the probability based on user selection
  probability <- reactive({
    if (input$prob_type == "less") {
      prob <- punif(input$x_value, min = input$a, max = input$b)
      explanation <- sprintf("P(X ≤ %.2f) = %.4f or %.2f%%", 
                            input$x_value, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "less", x = input$x_value))
      
    } else if (input$prob_type == "greater") {
      prob <- 1 - punif(input$x_value, min = input$a, max = input$b)
      explanation <- sprintf("P(X ≥ %.2f) = %.4f or %.2f%%", 
                            input$x_value, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "greater", x = input$x_value))
      
    } else if (input$prob_type == "between") {
      upper_prob <- punif(input$x_upper, min = input$a, max = input$b)
      lower_prob <- punif(input$x_lower, min = input$a, max = input$b)
      prob <- upper_prob - lower_prob
      
      explanation <- sprintf("P(%.2f ≤ X ≤ %.2f) = %.4f or %.2f%%", 
                            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 continuous uniform distribution plot
  output$distPlot <- renderPlot({
    # Create data frame for plotting the PDF
    x_range <- seq(input$a - 0.5 * (input$b - input$a), 
                   input$b + 0.5 * (input$b - input$a), 
                   length.out = 1000)
    
    pdf_values <- dunif(x_range, min = input$a, max = input$b)
    df <- data.frame(x = x_range, density = pdf_values)
    
    # Create base plot
    p <- ggplot(df, aes(x = x, y = density)) +
      geom_line(color = "darkgray", size = 1.2) +
      labs(x = "X", y = "probability density function") +
      theme_minimal() +
      theme(panel.grid.minor = element_blank()) +
      ylim(0, max(pdf_values) * 1.1)
    
    # Add shaded area based on selected probability type
    res <- probability()
    
    if (res$type == "less") {
      shade_x <- seq(input$a, res$x, length.out = 100)
      shade_y <- dunif(shade_x, min = input$a, max = input$b)
      shade_df <- data.frame(x = c(input$a, shade_x, res$x), 
                            y = c(0, shade_y, 0))
      
      p <- p + geom_polygon(data = shade_df, aes(x = x, y = y), 
                           fill = "#3F6BB6", alpha = 0.6)
      
    } else if (res$type == "greater") {
      shade_x <- seq(res$x, input$b, length.out = 100)
      shade_y <- dunif(shade_x, min = input$a, max = input$b)
      shade_df <- data.frame(x = c(res$x, shade_x, input$b), 
                            y = c(0, shade_y, 0))
      
      p <- p + geom_polygon(data = shade_df, aes(x = x, y = y), 
                           fill = "#3F6BB6", alpha = 0.6)
      
    } else if (res$type == "between") {
      shade_x <- seq(res$lower, res$upper, length.out = 100)
      shade_y <- dunif(shade_x, min = input$a, max = input$b)
      shade_df <- data.frame(x = c(res$lower, shade_x, res$upper), 
                            y = c(0, shade_y, 0))
      
      p <- p + geom_polygon(data = shade_df, aes(x = x, y = y), 
                           fill = "#3F6BB6", alpha = 0.6)
    }
    
    return(p)
  })
}

shinyApp(ui = ui, server = server)

Where to use: The continuous uniform distribution is used when all continuous values \(x\) in the interval \(a\) to \(b\) are equally likely. The random variable \(X\) represents the outcome.

Notation: \(X \sim \textrm{Uniform}(a,b)\) or \(X \sim U(a,b)\).

Parameters: Two real numbers \(a,b\), where

\(a\) is the minimum value of an outcome,
\(b\) is the maximum value of an outcome.

Quantity	Value	Notes
Mean	\(\mathbb{E}(X) = \dfrac{a+b}{2}\)
Variance	\(\mathbb{V}(X) = \dfrac{(b-a)^2}{12}\)
PDF	\(\mathbb{P}(X=x)=\begin{cases} \dfrac{1}{b-a} & \textsf{if } a \leq x \leq b \\0 & \textsf{otherwise}\end{cases}\)
CDF	\(\displaystyle\mathbb{P}(X\leq x)=\begin{cases} 0 & \textsf{if } x< a \\\dfrac{x-a}{b-a} & \textsf{if } a\leq x\leq b \\1 & \textsf{if } x>b \end{cases}\)

Example: A machine from Cantor’s Confectionery is programmed to chop long candy bars into pieces, each with a length between 30 millimetres to 50 millimetres. Due to variations in the machine, each continuous value between this interval is equally likely. This can be expressed as \(X \sim U(30,50)\). It means 30 is the minimum value and 50 is the maximum value, where all continuous values of \(X\) for \(30 \leq x \leq 50\) are equally likely.

Factsheet: Continuous uniform distribution

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