Exponential Distribution Calculator Formula

Understand the math behind the exponential distribution calculator. Each variable explained with a worked example.

Use the Exponential Distribution Calculator

Formulas Used

PDF f(x)

pdf = lambda * pow(e, -lambda * x)

CDF P(X <= x)

cdf = 1 - pow(e, -lambda * x)

P(X > x)

survival = pow(e, -lambda * x)

Mean (1/lambda)

mean_val = 1 / lambda

Variables

VariableDescriptionDefault
lambdaRate Parameter (lambda)0.5
xValue (x)3

How It Works

How to Calculate Exponential Distribution Probabilities

Formulas

PDF: f(x) = lambda * e^(-lambda*x) for x >= 0

CDF: P(X <= x) = 1 - e^(-lambda*x)

The exponential distribution models the time between events in a Poisson process. The parameter lambda is the rate (events per unit time). The mean waiting time is 1/lambda. It has the memoryless property: P(X > s+t | X > s) = P(X > t).

Worked Example

Buses arrive at a rate of 0.5 per minute. What is the probability of waiting at most 3 minutes?

lambda = 0.5x = 3
  1. 01lambda = 0.5, x = 3
  2. 02PDF: f(3) = 0.5 * e^(-0.5*3) = 0.5 * e^(-1.5) = 0.5 * 0.2231 = 0.1116
  3. 03CDF: P(X <= 3) = 1 - e^(-1.5) = 1 - 0.2231 = 0.7769
  4. 04P(wait > 3 min) = e^(-1.5) = 0.2231
  5. 05Mean waiting time = 1 / 0.5 = 2 minutes

Ready to run the numbers?

Open Exponential Distribution Calculator