Exponential Distribution

Distributions and Statistical Models

Calculate probabilities and statistics for an exponential distribution.

Examples

Explore some common scenarios for the exponential distribution.

Light Bulb Lifespan

standard

A light bulb has an average lifespan of 2000 hours. What is the probability it lasts for at least 2500 hours?

λ: 0.0005, x: 2500

Type: cdfGreaterOrEqual

Customer Service Calls

real-world

Customer service calls arrive at a rate of 2 per minute. What is the probability that the next call arrives in less than 30 seconds (0.5 minutes)?

λ: 2, x: 0.5

Type: cdfLessThan

Radioactive Decay

scientific

A radioactive particle decays at a rate of λ = 0.1 per second. What is the probability density at exactly 5 seconds?

λ: 0.1, x: 5

Type: pdf

Bus Arrival

simple

The time between bus arrivals at a stop is exponentially distributed with a mean of 10 minutes. What is the probability the next bus arrives in more than 15 minutes?

λ: 0.1, x: 15

Type: cdfGreaterThan

Other Titles
Understanding the Exponential Distribution
A Comprehensive Guide to a Key Probability Concept

What is the Exponential Distribution?

  • Core Definition
  • Key Properties
  • The Memoryless Property
The exponential distribution is a continuous probability distribution used to model the time we need to wait before a given event occurs. It is often used in reliability analysis and queuing theory. A key parameter of this distribution is the rate parameter (λ), which represents the average number of events per unit of time.
Mathematical Formulation
The Probability Density Function (PDF) is given by f(x; λ) = λe^(-λx) for x ≥ 0. The Cumulative Distribution Function (CDF) is F(x; λ) = 1 - e^(-λx). These formulas are the foundation for all calculations related to this distribution.

Step-by-Step Guide to Using the Calculator

  • Inputting Parameters
  • Selecting Calculation Type
  • Interpreting Results
1. Enter the Rate Parameter (λ)
This value represents the average rate of events. For example, if a machine fails 2 times per year on average, λ = 2.
2. Enter the Value of x
This is the specific time or value you are interested in. For example, to find the probability of the machine failing within 6 months (0.5 years), x = 0.5.
3. Choose Your Calculation
Select the desired probability from the dropdown: P(X < x), P(X ≤ x), P(X > x), P(X ≥ x), or the PDF value at x.
4. Analyze the Output
The calculator provides the calculated probability, along with key statistics like the mean, median, and variance, giving a full picture of the distribution.

Real-World Applications of Exponential Distribution

  • Reliability Engineering
  • Queuing Theory
  • Finance and Insurance
Product Lifespan
Engineers use the exponential distribution to predict the lifespan of electronic components. The rate parameter λ corresponds to the failure rate.
Customer Wait Times
Businesses model the time between customer arrivals at a service desk or call center. This helps in resource allocation and improving customer satisfaction.
Financial Modeling
In finance, it can be used to model the time between large market shocks or the time until a company defaults on its debt.

Common Misconceptions and Correct Methods

  • Confusing with Poisson Distribution
  • The 'Memoryless' Misconception
  • Rate vs. Mean
Exponential vs. Poisson
The Poisson distribution models the number of events in a fixed interval, while the exponential distribution models the time between those events. They are related but describe different aspects of the same process.
Understanding the Memoryless Property
The memoryless property states that the probability of an event occurring in the next interval is independent of how much time has already passed. For a machine that hasn't failed for 100 hours, the probability of it failing in the next hour is the same as it was for a new machine. This is a unique and powerful, but often misunderstood, feature.
Rate (λ) vs. Mean (1/λ)
It's easy to confuse the rate parameter with the mean. Remember that the mean time between events is the reciprocal of the rate (Mean = 1/λ). A high rate means events occur frequently, so the mean time between them is short.

Mathematical Derivation and Examples

  • Derivation of the Mean
  • Derivation of the Variance
  • Worked Example
Deriving the Mean
The expected value or mean (E[X]) is calculated by integrating x * f(x) from 0 to infinity. Using integration by parts, E[X] = ∫[0, ∞] xλe^(-λx) dx, which simplifies to 1/λ.
Deriving the Variance
The variance (Var(X)) is E[X²] - (E[X])². First, we find E[X²] = ∫[0, ∞] x²λe^(-λx) dx = 2/λ². Then, Var(X) = (2/λ²) - (1/λ)² = 1/λ².
Example: Call Center
Calls arrive at a center at a rate of λ = 4 calls per hour. We want to find the probability that the next call arrives within 15 minutes (0.25 hours). We use the CDF: P(X ≤ 0.25) = 1 - e^(-4 * 0.25) = 1 - e^(-1) ≈ 1 - 0.3679 = 0.6321. There is a 63.21% chance the next call will be within 15 minutes.