How do I choose the rate parameter lambda?

Lambda represents the average number of events per unit time. If you know the average time between events (the mean), then lambda = 1 / mean. For example, if customers arrive on average every 5 minutes, then lambda = 1/5 = 0.2 arrivals per minute. Lambda must be a positive number.

Exponential Distribution Calculator

Calculate probabilities, visualize the PDF and CDF curves, and explore properties of the exponential distribution. Enter the rate parameter λ (lambda) and a value x to get P(X ≤ x), P(X > x), P(a ≤ X ≤ b), mean, variance, median, and step-by-step solutions with interactive graphs.

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About Exponential Distribution Calculator

The Exponential Distribution Calculator computes probabilities, visualizes the probability density function (PDF) and cumulative distribution function (CDF), and displays distribution properties for the exponential distribution $X \sim \text{Exp}(\lambda)$. Enter the rate parameter $\lambda$ and a value $x$ to get $P(X \leq x)$, $P(X > x)$, or $P(a \leq X \leq b)$, along with step-by-step solutions, interactive graphs, and key statistics like the mean, variance, and median.

What Is the Exponential Distribution?

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process — a process where events occur continuously and independently at a constant average rate $\lambda$. It is defined by a single parameter $\lambda > 0$ (the rate parameter), and its probability density function (PDF) is:

$$f(x) = \lambda e^{-\lambda x}, \quad x \geq 0$$

The exponential distribution is widely used in reliability engineering, queueing theory, survival analysis, and telecommunications to model waiting times, lifetimes of components, and inter-arrival times.

Key Properties

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Memoryless

P(X > s+t | X > s) = P(X > t). The only continuous memoryless distribution.

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Mean = 1/λ

The expected waiting time is the reciprocal of the rate parameter.

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Poisson Link

If events follow Poisson(λ), time between events follows Exp(λ).

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Right-Skewed

Skewness = 2. Most values cluster near 0 with a long right tail.

Formulas

Property	Formula	Description
PDF	$f(x) = \lambda e^{-\lambda x}$	Probability density at x
CDF	$F(x) = 1 - e^{-\lambda x}$	Probability that X ≤ x
Survival	$S(x) = e^{-\lambda x}$	Probability that X > x
Mean	$\mu = \frac{1}{\lambda}$	Expected value
Variance	$\sigma^2 = \frac{1}{\lambda^2}$	Spread of the distribution
Median	$\frac{\ln 2}{\lambda}$	50th percentile
Mode	$0$	Most probable value
Skewness	$2$	Always right-skewed
Kurtosis	$6$	Excess kurtosis
MGF	$\frac{\lambda}{\lambda - t}$ for $t < \lambda$	Moment generating function

Real-World Applications

Field	What λ Represents	What X Models
Queueing Theory	Customer arrival rate	Time between customer arrivals
Reliability	Failure rate of component	Time until next failure
Telecommunications	Call arrival rate	Time between phone calls
Nuclear Physics	Decay rate	Time between radioactive decay events
Finance	Default rate	Time until loan default
Epidemiology	Infection rate	Time between infection events

Exponential vs. Poisson Distribution

The exponential and Poisson distributions are closely related but model different quantities:

Feature	Exponential	Poisson
Type	Continuous	Discrete
Models	Time between events	Number of events in interval
Parameter	λ (rate)	λ (rate × time)
Support	[0, ∞)	{0, 1, 2, ...}
Mean	1/λ	λ

How to Use the Exponential Distribution Calculator

Enter the rate parameter λ: This is the average number of events per unit time. For example, if buses arrive on average every 10 minutes, then λ = 1/10 = 0.1 buses per minute.
Select the probability type: Choose P(X ≤ x) for cumulative probability, P(X > x) for survival probability, or P(a ≤ X ≤ b) for range probability.
Enter the x value or range: For single-point probabilities, enter x. For range probabilities, enter both lower bound a and upper bound b.
Review the results: Examine the probability, interactive PDF and CDF graphs with shaded probability regions, distribution properties (mean, variance, median), and the complete step-by-step solution.

FAQ

What is the exponential distribution?

The exponential distribution is a continuous probability distribution that models the time between independent events occurring at a constant average rate. It is defined by a single rate parameter λ (lambda). The probability density function is f(x) = λe^(−λx) for x ≥ 0. It is commonly used to model waiting times, service times, and lifetimes of components.

What is the memoryless property of the exponential distribution?

The memoryless property means that the probability of waiting an additional t units of time is independent of how long you have already waited. Mathematically, P(X > s+t | X > s) = P(X > t). The exponential distribution is the only continuous distribution with this property. This makes it ideal for modeling processes where the future does not depend on the past, such as the time until the next phone call or the lifetime of an electronic component.

What is the relationship between the exponential and Poisson distributions?

If events occur according to a Poisson process with rate λ events per unit time, then the time between consecutive events follows an exponential distribution with the same rate parameter λ. The Poisson distribution counts the number of events in a fixed time interval, while the exponential distribution models the waiting time between events. They are two sides of the same coin.

How do I choose the rate parameter λ?

Lambda (λ) represents the average number of events per unit time. If you know the average time between events (the mean), then λ = 1/mean. For example, if customers arrive on average every 5 minutes, then λ = 1/5 = 0.2 arrivals per minute. If a machine fails on average every 100 hours, then λ = 1/100 = 0.01 failures per hour. Lambda must be a positive number.

Can the exponential distribution take negative values?

No, the exponential distribution is defined only for non-negative values (x ≥ 0). The probability density function is zero for any x less than 0. This makes intuitive sense because it typically models durations or waiting times, which cannot be negative.

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"Exponential Distribution Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-14

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Property	Formula	Description
PDF	\(f(x) = \lambda e^{-\lambda x}\)	Probability density at x
CDF	\(F(x) = 1 - e^{-\lambda x}\)	Probability that X ≤ x
Survival	\(S(x) = e^{-\lambda x}\)	Probability that X > x
Mean	\(\mu = \frac{1}{\lambda}\)	Expected value
Variance	\(\sigma^2 = \frac{1}{\lambda^2}\)	Spread of the distribution
Median	\(\frac{\ln 2}{\lambda}\)	50th percentile
Mode	\(0\)	Most probable value
Skewness	\(2\)	Always right-skewed
Kurtosis	\(6\)	Excess kurtosis
MGF	\(\frac{\lambda}{\lambda - t}\) for \(t < \lambda\)	Moment generating function

Exponential Distribution Calculator

About Exponential Distribution Calculator

What Is the Exponential Distribution?

Key Properties

Formulas

Real-World Applications

Exponential vs. Poisson Distribution

How to Use the Exponential Distribution Calculator

FAQ

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