Exponential Distribution Calculator

Exponential Distribution Calculator

Enter the rate parameter \( \lambda \) and a value \( x \) to compute the CDF and PDF.

Results

CDF at \( x \):

PDF at \( x \):

Expected Value:

Variance:

Random Sample:

Exponential Distribution: \(F(x;\lambda)=1-e^{-\lambda x}\), \(f(x;\lambda)=\lambda e^{-\lambda x}\), Expected value = \(1/\lambda\), Variance = \(1/\lambda^2\).

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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