StatsCalculators

Exponential Distribution

Created:October 20, 2024

Last Updated:April 9, 2025

This calculator helps you compute the probabilities of an exponential distribution given the rate parameter (λ). You can find the probability of waiting times or lifetimes being equal to, less than, greater than, or between specific values. The distribution chart shows the probability density function (PDF) of the exponential distribution, which models the time between events in a Poisson point process.

Calculator

Parameters

Rate Parameter (λ)

Mean waiting time = 1/λ

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

Click Calculate to view the distribution chart

Learn More

Exponential Distribution: Definition, Formula, and Applications

Exponential Distribution

Definition: The exponential distribution models the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Formula:The probability density function (PDF) and cumulative distribution function (CDF) are given by:

f(x) = \lambda e^{-\lambda x}, \quad x \geq 0

F(x) = 1 - e^{-\lambda x}, \quad x \geq 0

Where:

$\lambda$ is the rate parameter (average rate of events)
$x$ is the time between events

Properties

Mean (Expected Value): $E(X) = \frac{1}{\lambda}$
Variance: $\text{Var}(X) = \frac{1}{\lambda^2}$
Memoryless Property: $P(X > s + t | X > s) = P(X > t)$ . This means that if an event has not occurred for some time s, the probability of waiting an additional time t is the same as the probability of waiting time t from the start. This unique property means the distribution has no "memory" of past waiting time. For example, if a light bulb has been working for 10 hours, the probability it will work for one more hour is exactly the same as the probability a brand new bulb will work for one hour. Let's prove this mathematically: $P(X > s + t | X > s) = \frac{P(X > s + t)}{P(X > s)} = \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(X > t)$ This shows that waiting an additional time t after already waiting for time s has the same probability as waiting time t from the start.
Support: $[0, \infty)$
Right-skewed distribution
Decreasing failure rate

How to Calculate Exponential Distribution in R

R

library(tidyverse)

lambda <- 2
x <- 1.5
prob_density <- dexp(x, rate = lambda)  # 0.0995741367357279
prob_cumulative <- pexp(x, rate = lambda) # 0.950212931632136
print(paste0("P(X = ", x, "): ", prob_density))
print(paste0("P(X <= ", x, "): ", prob_cumulative))

# mean and variance
mean <- 1/lambda
variance <- 1/(lambda^2)
print(paste0("Mean: ", mean)) # 0.5
print(paste0("Variance: ", variance)) # 0.25

# plot pdf
set.seed(42)
random_nums <- rexp(1000, rate = lambda)
x_values <- seq(0, 5, by = 0.01)
pdf_values <- dexp(x_values, rate = lambda)

ggplot(data.frame(x = x_values, y = pdf_values), aes(x = x, y = y)) +
  geom_line(color = "blue", linewidth = 1) +
  labs(
    x = "x",
    y = "Probability Density",
    title = "Exponential Distribution PDF (λ = 2)"
  ) +
  theme_minimal()

# plot cdf
cdf_values <- pexp(x_values, rate = lambda)

ggplot(data.frame(x = x_values, y = cdf_values), aes(x = x, y = y)) +
  geom_line(color = "red", linewidth = 1) +
  labs(
    x = "x",
    y = "Cumulative Probability",
    title = "Exponential Distribution CDF (λ = 2)"
  ) +
  theme_minimal()

How to Calculate Exponential Distribution in Python

Python

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

lam = 2

# probabilities
x = 1.5
prob_density = stats.expon.pdf(x, scale=1/lam)
prob_cumulative = stats.expon.cdf(x, scale=1/lam)
print(f"P(X = {x}): {prob_density:.6f}")
print(f"P(X <= {x}): {prob_cumulative:.6f}")

# mean and variance
mean = 1/lam
variance = 1/(lam**2)
print(f"Mean: {mean:.6f}")
print(f"Variance: {variance:.6f}")

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