Factsheet: Discrete uniform distribution

Statistics

Author

Michelle Arnetta and Tom Coleman

Summary

A factsheet for the discrete uniform distribution.

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

ui <- page_fluid(
  title = "Discrete 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 = 1, step = 1),
        numericInput("b", "Maximum value (b):", value = 10, step = 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 = 1, max = 10, value = 5, step = 1)
        ),
        conditionalPanel(
          condition = "input.prob_type == 'between'",
          sliderInput("x_lower", "Lower bound (x):", min = 1, max = 10, value = 3, step = 1),
          sliderInput("x_upper", "Upper bound (y):", min = 1, max = 10, value = 7, step = 1)
        )
      )
    ),
    
    # Right column - Plot
    card(
      card_header("Discrete 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 or equal to a
  observe({
    if (input$b < input$a) {
      updateNumericInput(session, "b", value = input$a)
    }
  })
  
  # 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$a, 5), input$b))
    updateSliderInput(session, "x_lower", min = input$a, max = input$b, value = min(max(input$a, 3), input$b))
    updateSliderInput(session, "x_upper", min = input$a, max = input$b, value = min(max(input$a, 7), input$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$plot_title <- renderUI({
    title <- sprintf("DUnif(a = %d, b = %d)", input$a, input$b)
    tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
  })
  
  # Discrete uniform probability mass function
  ddunif <- function(x, min, max) {
    ifelse(x >= min & x <= max & x == round(x), 1/(max - min + 1), 0)
  }
  
  # Discrete uniform cumulative distribution function
  pdunif <- function(q, min, max) {
    ifelse(q < min, 0, 
           ifelse(q >= max, 1, 
                  (floor(q) - min + 1) / (max - min + 1)))
  }
  
  # Calculate the probability based on user selection
  probability <- reactive({
    if (input$prob_type == "less") {
      prob <- pdunif(input$x_value, input$a, input$b)
      explanation <- sprintf("P(X ≤ %d) = %.4f or %.2f%%", 
                            input$x_value, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "less", x = input$x_value))
      
    } else if (input$prob_type == "greater") {
      # For P(X ≥ x), we need 1 - P(X < x) = 1 - P(X ≤ x-1)
      if (input$x_value <= input$a) {
        prob <- 1  # P(X ≥ a) is always 1 for discrete uniform
      } else {
        prob <- 1 - pdunif(input$x_value - 1, input$a, input$b)
      }
      explanation <- sprintf("P(X ≥ %d) = %.4f or %.2f%%", 
                            input$x_value, prob, prob * 100)
      return(list(prob = prob, explanation = explanation, type = "greater", x = input$x_value))
      
    } else if (input$prob_type == "between") {
      if (input$x_lower == input$x_upper) {
        # Exact probability for a single value
        prob <- ddunif(input$x_lower, input$a, input$b)
      } else {
        # P(x_lower ≤ X ≤ x_upper) = P(X ≤ x_upper) - P(X < x_lower) = P(X ≤ x_upper) - P(X ≤ x_lower-1)
        upper_prob <- pdunif(input$x_upper, input$a, input$b)
        if (input$x_lower <= input$a) {
          lower_prob <- 0
        } else {
          lower_prob <- pdunif(input$x_lower - 1, input$a, input$b)
        }
        prob <- upper_prob - lower_prob
      }
      explanation <- sprintf("P(%d ≤ X ≤ %d) = %.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 discrete uniform distribution plot
  output$distPlot <- renderPlot({
    # Create data frame for plotting
    x_values <- input$a:input$b
    prob_mass <- rep(1/(input$b - input$a + 1), length(x_values))
    df <- data.frame(x = x_values, probability = prob_mass)
    
    # Create base plot
    p <- ggplot(df, aes(x = x, y = probability)) +
      geom_col(fill = "lightgray", color = "darkgray", alpha = 0.7) +
      labs(x = "X", y = "probability mass function") +
      theme_minimal() +
      theme(panel.grid.minor = element_blank()) +
      scale_x_continuous(breaks = x_values) +
      ylim(0, max(prob_mass) * 1.1)
    
    # Add shaded area based on selected probability type
    res <- probability()
    
    if (res$type == "less") {
      highlight_x <- input$a:res$x
      highlight_df <- df[df$x %in% highlight_x, ]
      
      p <- p + geom_col(data = highlight_df, aes(x = x, y = probability), 
                       fill = "#3F6BB6", color = "darkgray", alpha = 0.8)
      
    } else if (res$type == "greater") {
      highlight_x <- res$x:input$b
      highlight_df <- df[df$x %in% highlight_x, ]
      
      p <- p + geom_col(data = highlight_df, aes(x = x, y = probability), 
                       fill = "#3F6BB6", color = "darkgray", alpha = 0.8)
      
    } else if (res$type == "between") {
      highlight_x <- res$lower:res$upper
      highlight_df <- df[df$x %in% highlight_x, ]
      
      p <- p + geom_col(data = highlight_df, aes(x = x, y = probability), 
                       fill = "#3F6BB6", color = "darkgray", alpha = 0.8)
    }
    
    return(p)
  })
}

shinyApp(ui = ui, server = server)

Where to use: The discrete uniform distribution is used when all integer outcomes \(x\) in the interval \(a\) to \(b\) are equally likely. \(X\) is a random variable for integer outcomes \(x\) where for \(a \leq x \leq b\), and the probability of each outcome \(1/n\), where \(n = b - a + 1\).

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

Parameters: The numbers \(a,b\) are integers where

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

There are \(n\) outcomes in total, with \(n = b - a + 1\).

Quantity	Value	Notes
Mean	\(\mathbb{E}(X) = \dfrac{a+b}{2}.\)
Variance	\(\mathbb{V}(X) = \dfrac{n^2-1}{12}.\)
PMF	\(\mathbb{P}(X=x)=\frac{1}{n}\)
CDF	\(\mathbb{P}(X\leq x)= \begin{cases} 0 & \textsf{if } x \leq a \\\dfrac{\lfloor x \rfloor - a + 1}{n} & \textsf{if } a< x<b \\1 & \textsf{if } x \geq b \end{cases}\)	\(\lfloor x \rfloor\) is the floor function

Example: You roll a fair six-sided die, where all outcomes (\(1, 2, 3, 4, 5,\) and \(6\)) are equally likely. This can be expressed as \(X \sim U(1,6)\). It means \(1\) is the minimum value and \(6\) is the maximum value, where all discrete values of \(X\) for \(1 \leq x \leq 6\) are equally likely.

Factsheet: Discrete uniform distribution

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