Exponential Distribution Calculator
Calculate exponential distribution PDF, CDF, and P(X > x) with rate parameter λ. Statistics tool for mean, median, variance, and interactive PDF chart.
About This Calculator
The Exponential Distribution Calculator computes probabilities and statistics for the exponential distribution — a continuous probability distribution that models the time between events occurring independently at a constant average rate. Given the rate parameter λ (lambda) and a target value x, the calculator returns the probability density function (PDF), cumulative distribution function (CDF), tail probability P(X > x), mean, median, variance, and standard deviation. The interactive chart visualizes the full PDF curve.
The exponential distribution's PDF formula is f(x; λ) = λe^(-λx) for x ≥ 0. The CDF is F(x; λ) = 1 - e^(-λx), and the survival function P(X > x) = e^(-λx). The mean equals 1/λ, median equals ln(2)/λ, variance equals 1/λ², and standard deviation equals 1/λ. Its key property is being memoryless — the probability of an event in the next moment is independent of how long you have already waited.
The exponential distribution is the continuous analogue of the geometric distribution and is intimately linked to the Poisson distribution: if events follow a Poisson process with rate λ, the inter-arrival times are exponentially distributed with rate λ. It is widely used in queueing theory, reliability engineering, survival analysis, insurance modeling, and telecommunications.
Regional Notes
The exponential distribution is a universal mathematical concept used across all regions. In the United States, it is taught in AP Statistics and college-level probability courses. In India, it features in the curriculum for engineering, statistics, and commerce programs, often in the context of queueing theory and reliability. In the United Kingdom, it appears in A-level Further Statistics and undergraduate mathematics and operations research courses. The mathematical formulas and interpretation are identical worldwide.
Frequently Asked Questions
What is exponential distribution?
Exponential distribution is a continuous probability distribution that models the time between events occurring independently at a constant average rate. It is memoryless — the probability of an event occurring in the next hour is the same regardless of how long you have already waited. It is closely related to the Poisson distribution, which models the number of events in a fixed interval.
How do you calculate the PDF of exponential distribution?
The probability density function (PDF) of exponential distribution is f(x; λ) = λe^(-λx) for x ≥ 0, where λ is the rate parameter. The PDF describes the relative likelihood of observing a specific time interval x between consecutive events. The PDF is highest at x = 0 and decreases exponentially as x increases.
What is the CDF formula for exponential distribution?
The cumulative distribution function (CDF) of exponential distribution is P(X ≤ x) = 1 - e^(-λx). This gives the probability that a random variable X is less than or equal to a specific value x. The complementary probability P(X > x) = e^(-λx) is also known as the survival function or tail probability.
What is the mean and variance of exponential distribution?
The mean (expected value) of exponential distribution is 1/λ, where λ is the rate parameter. The median is ln(2)/λ ≈ 0.693/λ. The variance is 1/λ² and the standard deviation is 1/λ. For an exponential distribution, the standard deviation equals the mean — a unique property that highlights its high variability.
What is the memoryless property of exponential distribution?
The memoryless property means P(X > s + t | X > t) = P(X > s) — the probability of waiting another s units is the same regardless of how long you have already waited t. This makes exponential distribution the continuous analogue of the geometric distribution. It is the only continuous distribution with this property.
When is exponential distribution used in real-world applications?
Exponential distribution is widely used in reliability engineering to model time between failures, queueing theory to model inter-arrival times (customers, phone calls, buses), survival analysis, insurance claims modeling, radioactive decay, and telecommunications. It also appears in finance for modeling time between trades and in healthcare for modeling patient survival times.
What is the relationship between exponential and Poisson distribution?
Exponential and Poisson distributions are intimately related — if events follow a Poisson process with average rate λ per unit time, then: (1) the number of events in a fixed interval follows Poisson distribution with parameter λt, and (2) the time between consecutive events follows exponential distribution with rate parameter λ. The mean inter-arrival time is 1/λ.
How do you use the exponential distribution calculator?
To use the exponential distribution calculator: enter the rate parameter λ (lambda), which is the reciprocal of the average time between events. Then enter the value x for which you want to compute probabilities. The calculator instantly returns PDF, CDF, P(X > x), mean, median, variance, and standard deviation. The interactive chart shows the full PDF curve.