Calculator: Poisson distribution
Summary
A calculator to work out pmfs and cdfs for the Poisson distribution.
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library(shiny)
library(bslib)
library(ggplot2)
ui <- page_fluid(
title = "Poisson distribution calculator",
layout_columns(
col_widths = c(4, 8),
# Left column - Inputs
card(
card_header("Parameters"),
card_body(
numericInput("lambda", "Rate parameter (λ):", value = 5, min = 0.1, step = 0.1),
hr(),
radioButtons("prob_type", "Probability to calculate:",
choices = list("P(X = x)" = "exact",
"P(X ≤ x)" = "less",
"P(X ≥ x)" = "greater",
"P(x ≤ X ≤ y)" = "between"),
selected = "exact"),
conditionalPanel(
condition = "input.prob_type == 'exact'",
sliderInput("x_exact", "x value:", min = 0, max = 20, value = 5, step = 1)
),
conditionalPanel(
condition = "input.prob_type == 'less'",
sliderInput("x_less", "x value:", min = 0, max = 20, value = 5, step = 1)
),
conditionalPanel(
condition = "input.prob_type == 'greater'",
sliderInput("x_greater", "x value:", min = 0, max = 20, value = 5, step = 1)
),
conditionalPanel(
condition = "input.prob_type == 'between'",
sliderInput("x_lower", "Lower bound (x):", min = 0, max = 20, value = 3, step = 1),
sliderInput("x_upper", "Upper bound (y):", min = 0, max = 20, value = 7, step = 1)
)
)
),
# Right column - Plot
card(
card_header("Poisson 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 lambda changes, adjust the range of sliders
observe({
# Set a reasonable max value as 3*lambda or at least 10
max_x <- max(round(input$lambda * 3), 10)
updateSliderInput(session, "x_exact", max = max_x)
updateSliderInput(session, "x_less", max = max_x)
updateSliderInput(session, "x_greater", max = max_x)
updateSliderInput(session, "x_lower", max = max_x)
updateSliderInput(session, "x_upper", max = max_x)
})
# Ensure that x_upper is always greater than or equal to x_lower
observe({
if (input$x_upper < input$x_lower) {
updateSliderInput(session, "x_upper", value = input$x_lower)
}
})
# Display the plot title with distribution parameters
output$plot_title <- renderUI({
title <- sprintf("Poisson(λ = %.1f)", input$lambda)
tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
})
# Calculate the probability based on user selection
probability <- reactive({
if (input$prob_type == "exact") {
prob <- dpois(input$x_exact, lambda = input$lambda)
explanation <- sprintf("P(X = %d) = %.6f or %.4f%%",
input$x_exact, prob, prob * 100)
return(list(prob = prob, explanation = explanation, type = "exact", x = input$x_exact))
} else if (input$prob_type == "less") {
prob <- ppois(input$x_less, lambda = input$lambda)
explanation <- sprintf("P(X ≤ %d) = %.6f or %.4f%%",
input$x_less, prob, prob * 100)
return(list(prob = prob, explanation = explanation, type = "less", x = input$x_less))
} else if (input$prob_type == "greater") {
# For P(X ≥ x), we need 1 - P(X < x) = 1 - P(X ≤ x-1)
if (input$x_greater == 0) {
prob <- 1 # P(X ≥ 0) is always 1
} else {
prob <- 1 - ppois(input$x_greater - 1, lambda = input$lambda)
}
explanation <- sprintf("P(X ≥ %d) = %.6f or %.4f%%",
input$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) {
# Exact probability for a single value
prob <- dpois(input$x_lower, lambda = input$lambda)
} 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 <- ppois(input$x_upper, lambda = input$lambda)
if (input$x_lower == 0) {
lower_prob <- 0
} else {
lower_prob <- ppois(input$x_lower - 1, lambda = input$lambda)
}
prob <- upper_prob - lower_prob
}
explanation <- sprintf("P(%d ≤ X ≤ %d) = %.6f or %.4f%%",
input$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$explanation)
})
# Generate the Poisson distribution plot
output$distPlot <- renderPlot({
# Determine the range for the x-axis
lambda <- input$lambda
max_x <- max(round(lambda * 3), 10)
# Create data frame for plotting
x_values <- 0:max_x
prob_mass <- dpois(x_values, lambda = lambda)
df <- data.frame(x = x_values, probability = prob_mass)
# Create base plot
p <- ggplot(df, aes(x = x, y = probability)) +
geom_col(fill = "lightgray", color = "darkgray", alpha = 0.7) +
labs(x = "number of events (X)", y = "probability mass function") +
theme_minimal() +
theme(panel.grid.minor = element_blank()) +
scale_x_continuous(breaks = seq(0, max_x, by = ifelse(max_x > 20, 2, 1)))
# Add shaded area based on selected probability type
res <- probability()
if (res$type == "exact") {
highlight_x <- res$x
highlight_df <- df[df$x == highlight_x, ]
p <- p + geom_col(data = highlight_df, aes(x = x, y = probability),
fill = "#3F6BB6", color = "darkgray", alpha = 0.8)
} else if (res$type == "less") {
highlight_x <- 0:res$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$x:max_x
highlight_df <- df[df$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$type == "between") {
highlight_x <- res$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)Further reading
[This interactive element appears in Overview: Probability distributions. Please click this link to go to the guide.]
Version history
v1.0: initial version created 12/24 by Ellie Trace as part of a University of St Andrews VIP project.
- v1.1: updated to R Shiny interface by tdhc 04/25.