Exponential Distribution Calculator - Probability & Statistics
Calculate exponential distribution probabilities, mean, variance, and cumulative probabilities using the rate parameter λ
Distribution Parameters
Results
What is an Exponential Distribution Calculator?
An Exponential Distribution Calculator computes probabilities and statistical properties for the exponential distribution, which models the time between events in a Poisson process.
This calculator is used for:
- Reliability Engineering - Component lifetime analysis
- Queuing Theory - Customer arrival and service times
- Physics - Radioactive decay modeling
- Finance - Time between transactions or defaults
To work with continuous probability distributions for bounded variables, explore our Beta Distribution Calculator to model probabilities between 0 and 1 for Bayesian analysis.
To understand discrete probability distributions, check out our Geometric Distribution Calculator to find the probability of trials until the first success.
To forecast values using exponential models, visit our Exponential Growth Prediction Calculator to predict future growth in finance, biology, and business.
To calculate confidence intervals for your time-based estimates, use our Confidence Interval Calculator to determine statistical ranges for population parameters.
How Exponential Distribution Works
The exponential distribution uses these formulas:
Where:
- λ (lambda) = Rate parameter (> 0)
- x = Time or value (≥ 0)
- e = Euler's number (≈ 2.71828)
Key Concepts Explained
Rate Parameter (λ)
Represents the average number of events per unit time. Higher λ means events occur more frequently.
Memoryless Property
The probability of an event in the next interval doesn't depend on how much time has already passed. Unique to exponential distribution.
Relationship to Poisson
If events follow a Poisson process with rate λ, then the time between events follows an exponential distribution with the same λ.
How to Use This Calculator
Enter Rate Parameter (λ)
Input the rate parameter (must be positive, e.g., 0.5 events per unit time)
Enter Value (x)
Input the time or value at which to calculate probabilities
View Results
Review probabilities, PDF value, mean, variance, and standard deviation
Benefits of This Calculator
- Instant Results - Immediate probability calculations
- Complete Analysis - PDF, CDF, and complementary probabilities
- Statistical Measures - Mean, variance, standard deviation
- Accurate Formulas - Uses standard exponential formulas
- Reliability Analysis - Perfect for lifetime modeling
- Educational Tool - Great for statistics and engineering students
Important Considerations
- Rate Parameter - λ must be positive; represents event frequency
- Time Scale - Ensure λ and x use consistent time units
- Non-negative Values - x must be ≥ 0 (time cannot be negative)
- Constant Rate - Assumes constant event rate over time
- Independence - Events must occur independently
- Right Skewed - Exponential distribution is always right-skewed
Frequently Asked Questions
What is an exponential distribution?
The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process. It describes the time until an event occurs, characterized by a constant rate parameter λ (lambda).
What is the formula for exponential distribution?
The probability density function is f(x) = λe^(-λx) for x ≥ 0. The cumulative distribution function is P(X ≤ x) = 1 - e^(-λx).
What is the mean of an exponential distribution?
The mean of an exponential distribution is μ = 1/λ, where λ is the rate parameter.
What is the variance of an exponential distribution?
The variance of an exponential distribution is σ² = 1/λ², where λ is the rate parameter. The standard deviation is σ = 1/λ.
When is exponential distribution used?
Exponential distribution is used for modeling waiting times, time between arrivals, component lifetimes, radioactive decay, and any process where events occur continuously and independently at a constant average rate.
What is the memoryless property of exponential distribution?
The exponential distribution has the memoryless property: the probability of an event occurring in the next interval is independent of how much time has already elapsed. This is unique to the exponential distribution among continuous distributions.