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Exponential Distribution Calculators

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Exponential Distribution Calculator

Exponential Distribution Calculator

Enter the rate parameter λ and a value x to compute the CDF and PDF.

Results

CDF at x:

PDF at x:

Expected Value:

Variance:

Random Sample:

Exponential Distribution: F(x;λ)=1−e−λx, f(x;λ)=λe−λx, Expected value = 1/λ, Variance = 1/λ2.

Exponential Distribution

Exponential Distribution

The exponential distribution is widely used to model the time between events. It is defined by the parameter λ (>0).

Probability Density Function (PDF)

The PDF of an exponential random variable X (with parameter λ) is given by:

        fX(x) = λe-λx, for x > 0
        fX(x) = 0, otherwise

Cumulative Distribution Function (CDF)

The CDF of X is:

        FX(x) = (1 - e-λx), for x ≥ 0
        FX(x) = 0, otherwise

Mean and Variance

The expected value (mean) and variance of X are:

        E[X] = 1/λ

        Var(X) = 1/λ²

Intuition and Properties

The exponential distribution can be seen as the continuous version of the geometric distribution. It represents waiting times, like waiting for customers to arrive at a store.

A key property is the “memoryless” feature:

        P(X > x + a | X > a) = P(X > x), for a, x ≥ 0

This means the probability of waiting an additional time x doesn’t depend on how long you have already waited.

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