Exponential Distribution

Basic Concepts

The exponential distribution can be used to determine the probability that it will take a given number of trials to arrive at the first success in a Poisson distribution; i.e. it describes the inter-arrival times in a Poisson process. It is the continuous counterpart to the geometric distribution, and it too is memoryless.

Definition 1: The exponential distribution has the probability density function (pdf) given by

f(x) = λe-λx

for x ≥ 0. Lambda is called the rate parameter and λ > 0.

The cumulative distribution function (cdf) is

F(x) = 1 – e-λx

The inverse cumulative distribution function is

F-1(p) = – ln(1–p)/λ

Worksheet Functions

Excel Function: Excel provides the following function for the exponential distribution:

EXPON.DIST(x, λ, cum) = the pdf of the exponential function f(x) when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

This function is not available in versions of Excel prior to Excel 2010. Instead, these versions of Excel use the equivalent function EXPONDIST.

Observation: The exponential distribution is equivalent to the gamma distribution with α = 1 and β = 1/λ. Thus, EXPON.DIST(x, λ, cum) = GAMMA.DIST(x, 1, 1/λ, cum).

There is no EXPON.INV(p, λ) function in Excel, but GAMMA.INV(p, 1, 1/λ) or –LN(1–p)/λ or the following Real Statistics function can be used instead.

Real Statistics Function: The Real Statistics Resource Pack supplies the following function.

EXPON_INV(p, λ) = the inverse of the exponential distribution at p

Graph

Figure 1 shows a graph of the pdf of the exponential distribution for λ = 1, 2 and 3.

Exponential distribution chart

Figure 1 – Pdf of exponential distribution

Properties

Key statistical properties are:

  • Mean = 1 / λ
  • Median = ln 2/λ
  • Mode = 0
  • Range = [0, ∞)
  • Variance = 1 / λ2
  • Skewness = 2
  • Kurtosis = 6

If λ is a constant representing the average number of random events that occur in a fixed time interval, then the probability that the first such event will occur in less than x time is given by the cumulative exponential distribution function F(x).

Property 1: An exponential distribution is memoryless

Property 2: If x has a Poisson distribution with mean λ, then the time between events follows an exponential distribution with mean 1/ λ.

Click here for the proofs of these two properties.

Related Distributions

  • If x ∼ Laplace(μ, β), then |x – μ| ∼ Exp(1/β)
  • If x ∼ Pareto(α, 1), then ln x ∼ Exp(α)
  • If x ∼ Exp(λ), then x ∼ Weibull(1/λ, 1)
  • If x ∼ Exp(λ), then x2 ∼ Weibull(1/λ2, 1/2)
  • If x ∼ Exp(λ), then e-x ∼ Beta(λ, 1)
  • If x ∼ Exp(λ), then kex ∼ Pareto(λ, k)

Examples

Click here for examples based on the exponential distribution.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Wikipedia (2012) Exponential distribution
https://en.wikipedia.org/wiki/Exponential_distribution

18 thoughts on “Exponential Distribution”

  1. Hi Charles,

    This is a massive help. I think its exactly what I’ve been looking for.

    I want to conduct a simulation and I think part of it should be from the exponential dist.
    I’m simulating a utility over time. The utility value will be on the beta dist.
    The utility is inferred to be constant over time, so I plan to draw from the exponential dist on excel for the time period.

    I will use: LN(1 – RAND()) / -λ

    Do you think this approach sounds correct?
    Would you have any references for this?

    Many thanks
    B

    Reply

  2. Instead of using the Gamma.inv function, we can use the following formula:

    = LN(1 – RAND()) / -λ

    for simulation for an F(x) = RAND(), when we want to find the ‘x’.

    Reply

  6. Dear frinds
    Give me help
    I need a real data set in any field that have exponential distribution.
    size of data should be at least 300

    Reply

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