27  Exponential Distribution

The Exponential distribution answers a single practical question: how long until the next event? Whenever events occur at a constant average rate — equipment failures, customer arrivals, radioactive decays — the gap between them follows an Exponential distribution.

Formally, the random variate \(X\) defined for the range \(X \in [0, \infty)\), is said to have an Exponential Distribution (i.e. \(X \sim \text{Exp}(\lambda)\)) with rate parameter \(\lambda > 0\).

Parameterization note. Some texts use the mean parameter \(\beta = 1/\lambda\) (so that \(\text{E}(X) = \beta\) directly). Exponential functions in base R (dexp, pexp, qexp, rexp) use rate; to work with scale \(\beta = 1/\lambda\), pass rate = 1/beta. Throughout this chapter we use the rate parameterization \(\lambda\).

27.1 Probability Density Function

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

The figure below shows examples of the Exponential Probability Density Function for different values of \(\lambda\).

Code 
par(mfrow = c(2, 2))
x <- seq(0, 6, length = 500)

plot(x, dexp(x, rate = 0.5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(lambda == 0.5))

plot(x, dexp(x, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(lambda == 1))

plot(x, dexp(x, rate = 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(lambda == 2))

plot(x, dexp(x, rate = 5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(lambda == 5))

par(mfrow = c(1, 1))
Figure 27.1: Exponential Probability Density Function for various values of lambda

27.2 Purpose

The Exponential distribution is used to model waiting times and lifetimes when the rate of events is constant over time — equivalently, when the probability of an event in the next small interval does not depend on how much time has already elapsed (the memoryless property). Common applications include:

  • Time between successive events in a Poisson process (interarrival times)
  • Time to failure in reliability engineering (electronic components, machinery)
  • Duration of telephone calls and customer service interactions
  • Response and service times in queuing models
  • Radioactive decay times in nuclear physics

Relation to the discrete setting. The Exponential distribution is the continuous analog of the Geometric distribution — both share the memoryless property. Where the Geometric counts the number of discrete Bernoulli trials until the first success, the Exponential measures the continuous waiting time until the first event in a Poisson process. In both cases, past history provides no information about future waiting: the process always “starts fresh.” Both distributions have a coefficient of variation equal to 1.

27.3 Distribution Function

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

The figure below shows the Exponential Distribution Function for \(\lambda = 1\).

Code 
x <- seq(0, 6, length = 500)
plot(x, pexp(x, rate = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Exponential Distribution Function",
     sub = expression(lambda == 1))
Figure 27.2: Exponential Distribution Function (lambda = 1)

27.4 Moment Generating Function

\[ M_X(t) = \frac{\lambda}{\lambda - t}, \quad t < \lambda \]

27.5 1st Uncentered Moment

\[ \mu_1' = \frac{1}{\lambda} \]

27.6 2nd Uncentered Moment

\[ \mu_2' = \frac{2}{\lambda^2} \]

27.7 3rd Uncentered Moment

\[ \mu_3' = \frac{6}{\lambda^3} \]

27.8 4th Uncentered Moment

\[ \mu_4' = \frac{24}{\lambda^4} \]

27.9 2nd Centered Moment

\[ \mu_2 = \frac{1}{\lambda^2} \]

27.10 3rd Centered Moment

\[ \mu_3 = \frac{2}{\lambda^3} \]

27.11 4th Centered Moment

\[ \mu_4 = \frac{9}{\lambda^4} \]

27.12 Expected Value

\[ \text{E}(X) = \frac{1}{\lambda} \]

27.13 Variance

\[ \text{V}(X) = \frac{1}{\lambda^2} \]

27.14 Median

\[ \text{Med}(X) = \frac{\ln 2}{\lambda} \]

27.15 Mode

\[ \text{Mo}(X) = 0 \]

The density is strictly decreasing on \([0, \infty)\), so the mode is always at the left boundary of the support.

27.16 Coefficient of Skewness

\[ g_1 = 2 \]

The Exponential distribution is always positively skewed, with skewness equal to 2 regardless of \(\lambda\).

27.17 Coefficient of Kurtosis

\[ g_2 = 9 \]

The excess kurtosis is \(g_2 - 3 = 6\), indicating heavier tails than the Normal distribution.

27.18 Parameter Estimation

The maximum likelihood estimator of the rate parameter is the reciprocal of the sample mean:

\[ \hat{\lambda} = \frac{1}{\bar{x}} = \frac{n}{\sum_{i=1}^{n} x_i} \]

Note that \(1/\hat{\lambda} = \bar{x}\) is an unbiased estimator of \(1/\lambda = \text{E}(X)\). However, \(\hat{\lambda}\) itself is biased for \(\lambda\); the minimum-variance unbiased estimator of \(\lambda\) is \((n-1)/(n\bar{x})\).

27.19 R Module

27.19.1 RFC

The Exponential Distribution module is available in RFC under the menu “Distributions / Exponential Distribution”.

27.19.3 R Code

The following code demonstrates Exponential probability calculations:

# Probability density function: f(x)
dexp(x = 100, rate = 0.005)

# Distribution function: P(X <= x)
pexp(q = 100, rate = 0.005)

# Quantile function: find x such that P(X <= x) = p
qexp(p = 0.5, rate = 0.005)

# Generate random Exponential numbers
set.seed(42)
rexp(n = 10, rate = 0.005)
[1] 0.003032653
[1] 0.3934693
[1] 138.6294
 [1]  39.66736 132.17905  56.69821   7.63838  94.63533 292.72543  62.79692
 [8]  82.02591 238.31956 142.97250

To fit an Exponential distribution to observed data:

library(MASS)

# Example: time-to-failure data (hours)
failures <- c(312, 87, 523, 145, 210, 403, 67, 289, 178, 356,
              94, 441, 233, 512, 162, 378, 55, 297, 189, 421)

# Maximum likelihood estimation
fit <- fitdistr(failures, "exponential")
print(fit)

# Compare with reciprocal of sample mean
cat("\nSample mean:", mean(failures), "\n")
cat("MLE rate (1/mean):", 1/mean(failures), "\n")
       rate    
  0.0037369208 
 (0.0008356009)

Sample mean: 267.6 
MLE rate (1/mean): 0.003736921

27.20 Example

A data center monitors server uptime before hardware failure. Based on historical records, the mean time between failures (MTBF) is 200 hours, corresponding to a failure rate of \(\lambda = 1/200 = 0.005\) failures per hour. We model the time to failure as \(X \sim \text{Exp}(0.005)\).

lambda <- 0.005

# P(X <= 100): server fails within the first 100 hours
cat("P(failure within 100 h):", pexp(100, rate = lambda), "\n")

# P(X > 500): server survives beyond 500 hours
cat("P(survives beyond 500 h):", 1 - pexp(500, rate = lambda), "\n")

# Median time to failure
cat("Median time to failure (h):", qexp(0.5, rate = lambda), "\n")

# 90th percentile: 90% of failures occur before this time
cat("90th percentile (h):", qexp(0.9, rate = lambda), "\n")
P(failure within 100 h): 0.3934693 
P(survives beyond 500 h): 0.082085 
Median time to failure (h): 138.6294 
90th percentile (h): 460.517

You can reproduce this exact scenario with the preconfigured Exponential app:

Interactive Shiny app (click to load).
Open in new tab

27.21 Random Number Generator

Exponential random variates can be generated from uniform random numbers via the inverse-CDF (probability integral transform). Since \(F(x) = 1 - e^{-\lambda x}\), setting \(U = F(X)\) and solving for \(X\) gives:

\[ X = -\frac{\ln(1 - U)}{\lambda} \sim \text{Exp}(\lambda) \quad \text{when } U \sim \text{U}(0,1) \]

Because \(1 - U\) is also uniform on \((0,1)\), the equivalent form \(X = -\ln(U)/\lambda\) is commonly used.

set.seed(123)
n     <- 1000
lambda <- 1

# Inverse-transform method
u <- runif(n)
x_inv <- -log(u) / lambda

# Built-in function
x_rexp <- rexp(n, rate = lambda)

cat("Inverse-transform: mean =", round(mean(x_inv), 4),
    "  var =", round(var(x_inv), 4), "\n")
cat("rexp():            mean =", round(mean(x_rexp), 4),
    "  var =", round(var(x_rexp), 4), "\n")
cat("Theoretical:       mean =", 1/lambda,
    "  var =", 1/lambda^2, "\n")
Inverse-transform: mean = 1.0064   var = 1.016 
rexp():            mean = 1.0331   var = 1.0425 
Theoretical:       mean = 1   var = 1
Code 
set.seed(123)
x <- rexp(1000, rate = 1)
hist(x, breaks = 30, col = "steelblue", freq = FALSE,
     xlab = "x", main = "Exponential Random Numbers (n = 1000, lambda = 1)")
curve(dexp(x, rate = 1), add = TRUE, col = "red", lwd = 2)
legend("topright", legend = "Theoretical density", col = "red", lwd = 2)
Figure 27.3: Histogram of simulated Exponential random numbers (n = 1000, lambda = 1)
Interactive Shiny app (click to load).
Open in new tab

27.22 Property 1: Memoryless Property

The Exponential distribution is the only continuous distribution with the memoryless property:

\[ \text{P}(X > s + t \mid X > s) = \text{P}(X > t) \quad \text{for all } s, t \geq 0 \]

Intuitively, if a server has already been running for \(s\) hours without failure, the distribution of remaining uptime is identical to the original distribution. Past survival provides no information about future failure.

27.23 Property 2: Minimum of Independent Exponentials

If \(X_1 \sim \text{Exp}(\lambda_1)\) and \(X_2 \sim \text{Exp}(\lambda_2)\) are independent, then their minimum is also exponentially distributed:

\[ \min(X_1, X_2) \sim \text{Exp}(\lambda_1 + \lambda_2) \]

More generally, the minimum of \(n\) independent exponentials with rates \(\lambda_1, \ldots, \lambda_n\) follows \(\text{Exp}(\lambda_1 + \cdots + \lambda_n)\).

27.24 Property 3: Coefficient of Variation Equals 1

The coefficient of variation (CV = standard deviation / mean) is always exactly 1:

\[ \text{CV} = \frac{\sqrt{1/\lambda^2}}{1/\lambda} = 1 \]

This holds for every value of \(\lambda\). An observed CV substantially different from 1 suggests the Exponential model may not be appropriate for the data.