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Exponential Distribution

Formula

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

Plot

fn: exp(-x)
xmin: 0
xmax: 6
ymin: 0
ymax: 1.1
height: 280
title: Exponential PDF (lambda=1)

Parameters

  • \(\lambda>0\): rate
  • \(x\ge 0\): waiting time

What it means

Models waiting times between events in a Poisson process.

What it's used for

  • Time-to-event modeling with constant hazard.
  • Queueing and reliability basics.

Key properties

  • Memoryless continuous distribution.
  • Mean \(1/\lambda\), variance \(1/\lambda^2\).

Common gotchas

  • Rate \(\lambda\) vs scale \(1/\lambda\) parameterizations are both used.
  • Not appropriate when hazard changes over time.

Example

If \(\lambda=2\) per hour, expected waiting time is \(0.5\) hours.

How to Compute (Pseudocode)

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity

  • Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
  • Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
  • Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families

See also