Factsheet: Gamma distribution
Statistics
Summary
A factsheet for the gamma distribution.
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library(shiny)
library(bslib)
library(ggplot2)
ui <- page_fluid(
title = "Gamma distribution calculator",
layout_columns(
col_widths = c(4, 8),
# Left column - Inputs
card(
card_header("Parameters"),
card_body(
numericInput("shape", "Shape parameter (α):", value = 2, min = 0.01, step = 0.1),
numericInput("scale", "Scale parameter (θ):", value = 1, 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 = 15, value = 2, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'greater'",
sliderInput("x_greater", "x value:", min = 0, max = 15, value = 2, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'between'",
sliderInput("x_lower", "Lower bound (x):", min = 0, max = 15, value = 1, step = 0.1),
sliderInput("x_upper", "Upper bound (y):", min = 0, max = 15, value = 5, step = 0.1)
)
)
),
# Right column - Plot
card(
card_header("Gamma 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 parameters change, adjust the range of sliders
observe({
# For gamma distribution, adjust slider based on parameters
shape <- input$shape
scale <- input$scale
# Use a heuristic to determine a reasonable upper bound
# This captures most of the meaningful density
max_x <- min(qgamma(0.995, shape = shape, scale = scale), 100)
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("Gamma(α = %.2f, θ = %.2f)", input$shape, input$scale)
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 <- pgamma(input$x_less, shape = input$shape, scale = input$scale)
explanation <- sprintf("P(X ≤ %.1f) = %.6f or %.4f%%",
input$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 - pgamma(input$x_greater, shape = input$shape, scale = input$scale)
explanation <- sprintf("P(X ≥ %.1f) = %.6f or %.4f%%",
input$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 <- pgamma(input$x_upper, shape = input$shape, scale = input$scale)
lower_prob <- pgamma(input$x_lower, shape = input$shape, scale = input$scale)
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 gamma distribution plot
output$distPlot <- renderPlot({
# Get parameters
shape <- input$shape
scale <- input$scale
# Determine a reasonable max for x-axis based on parameters
max_x <- min(qgamma(0.995, shape = shape, scale = scale), 100)
# Create data frame for plotting
x_values <- seq(0.01, max_x, length.out = 500) # Avoid x=0 for some shape values
density_values <- dgamma(x_values, shape = shape, scale = scale)
plot_df <- data.frame(x = x_values, density = density_values)
# Create base plot
p <- ggplot(plot_df, aes(x = x, y = density)) +
geom_line(size = 1, color = "darkgray") +
labs(x = "X", y = "Probability Density Function") +
theme_minimal() +
theme(panel.grid.minor = element_blank()) +
xlim(0, max_x)
# Add shaded area based on selected probability type
res <- probability()
if (res$type == "less") {
# Create data for the filled area
fill_x <- seq(0.01, res$x, length.out = 200)
fill_y <- dgamma(fill_x, shape = shape, scale = scale)
fill_df <- data.frame(x = fill_x, density = fill_y)
p <- p + geom_area(data = fill_df, aes(x = x, y = density),
fill = "#3F6BB6", alpha = 0.6)
} else if (res$type == "greater") {
# Create data for the filled area
fill_x <- seq(res$x, max_x, length.out = 200)
fill_y <- dgamma(fill_x, shape = shape, scale = scale)
fill_df <- data.frame(x = fill_x, density = fill_y)
p <- p + geom_area(data = fill_df, aes(x = x, y = density),
fill = "#3F6BB6", alpha = 0.6)
} else if (res$type == "between") {
# Create data for the filled area
fill_x <- seq(res$lower, res$upper, length.out = 200)
fill_y <- dgamma(fill_x, shape = shape, scale = scale)
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 gamma distribution generalizes the exponential distribution, allowing for greater or lesser variance. It is used to model positive continuous random variables that have skewed distributions.
Notation: \(X \sim \textrm{Gamma}(\alpha,\theta)\) or \(X \sim \textrm{Gam}(\alpha,\theta)\)
Parameters: Two real numbers \(\alpha\) and \(\theta\), which are related to the mean \(\mu\) and variance \(\sigma^2\):
- \(\alpha = \frac{\mu^2}{\sigma^2}\) (shape parameter)
- \(\theta=\frac{\sigma^2}{\mu}\) (scale parameter)
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = \alpha\theta\) | |
| Variance | \(\mathbb{V}(X) = \alpha\theta^2\) | |
| \(\mathbb{P}(X=x)=\dfrac{x^{\alpha-1}\exp\left(-\frac{x}{\theta}\right)}{\Gamma(\alpha)\theta^{\alpha}}\) | \(\Gamma(x)\) the gamma function of \(x\) | |
| CDF | \(\mathbb{P}(X \leq x)=\dfrac{\textrm{Gam}\left(\alpha,\frac{x}{\theta}\right)}{\Gamma(\alpha)}\) | \(\textrm{Gam}(\alpha,\theta)\) is the PDF of the gamma distribution |
Example: You collect historical data on the time to failure of a machine from Cantor’s Confectionery. The mean is 83 days and the variance is 50.3. You can then use this to estimate the shape and scale parameters of the gamma distribution:
\(\alpha = \frac{83^2}{50.3} = 136.958250497 \approx 137\)
\(\theta = \frac{50.3}{83} = 0.60602409638 \approx 0.61\)
The distribution can be expressed as \(X \sim \textrm{Gam}(137,0.61)\), where the shape parameter is 137 and the scale parameter is 0.61.
Further reading
Version history
v1.0: initial version created 04/25 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.
- v1.1: moved to factsheet form and populated with material from Overview: Probability distributions by tdhc.