What is the exponential distribution used for?

The exponential distribution models the time between independent events that occur at a constant average rate, such as time to failure of a component, waiting time between arrivals in a Poisson process, or time until the next call in a call center.

What are the parameters of the exponential distribution?

The exponential distribution is usually parameterized either by the rate λ (lambda) > 0, which is the average number of events per unit time, or by the mean μ = 1/λ, which is the average waiting time. This calculator lets you work with either parameterization.

What is the memoryless property of the exponential distribution?

The exponential distribution is memoryless, meaning that P(X > s + t | X > s) = P(X > t). In words, the remaining waiting time does not depend on how much time has already elapsed. This is unique among continuous distributions and is a key reason the exponential is used for modeling Poisson process waiting times.

How do I compute P(X ≤ x) and P(X > x) for an exponential distribution?

For an exponential distribution with rate λ, the CDF is F(x) = 1 − e^{−λx} for x ≥ 0, so P(X ≤ x) = 1 − e^{−λx}. The survival function is S(x) = P(X > x) = e^{−λx}. This calculator computes both values for you and shows the formulas.

How do I find quantiles or percentiles of the exponential distribution?

For 0 < p < 1, the p-quantile of an exponential distribution with rate λ is Q(p) = −ln(1 − p) / λ. For example, the median is Q(0.5) = ln(2)/λ. Enter p in the calculator to get the corresponding quantile.

Exponential Distribution Calculator

Q: How do I find quantiles or percentiles of the exponential distribution?

For 0 < p < 1, the p-quantile of an exponential distribution with rate λ is Q(p) = −ln(1 − p) / λ. For example, the median is Q(0.5) = ln(2)/λ. Enter p in the calculator to get the corresponding quantile.

Compute PDF, CDF, survival, quantiles, and random samples for the exponential distribution. Supports both rate (λ) and mean (μ) parameterizations, with instant plots and formulas.

1. Set parameters

Parameterization

Rate λ (events per unit time) Mean μ (average waiting time)

Rate λ > 0

Example: λ = 0.5 means on average 0.5 events per unit time (mean waiting time μ = 2).

Point x ≥ 0

Used for PDF f(x), CDF F(x), and survival S(x) = P(X > x).

Probability p (0 < p < 1) for quantile

Quantile Q(p) solves P(X ≤ Q(p)) = p. For example p = 0.5 gives the median.

Interval start a ≥ 0

Interval end b ≥ a

Used for P(a ≤ X ≤ b).

Memoryless t ≥ 0

Used for P(X > s + t | X > s) = P(X > t).

All results update instantly when you change inputs.

2. Results

Point probabilities PDF / CDF

PDF f(x): –
CDF F(x) = P(X ≤ x): –
Survival S(x) = P(X > x): –

Interval & quantile Tail probs

P(a ≤ X ≤ b): –
Quantile Q(p): –
Median (p = 0.5): –

Moments Mean / Var

Mean E[X]: –
Variance Var(X): –
Std. deviation σ: –

Memoryless property Unique

P(X > s + t | X > s) –

P(X > t) –

For an exponential distribution these two values are equal (up to rounding).

3. Visualize the distribution

Drag the slider or edit the parameter to see how the exponential PDF and CDF change.

Interactive rate λ for plots

0.1 λ = 0.5 3.0

PDF f(x) = λ e^−λx

CDF F(x) = 1 − e^−λx

Exponential distribution: definition and formulas

The exponential distribution is a continuous probability distribution on [0, \infty) commonly used to model waiting times between independent events that occur at a constant average rate (a Poisson process).

Parameterizations

There are two equivalent ways to specify an exponential distribution:

Rate form: parameter λ > 0 (events per unit time). We write \(X \sim \text{Exp}(\lambda)\).
Mean form: parameter μ > 0 (average waiting time). Here μ = 1/λ.

PDF (probability density function)

\[ f(x) = \begin{cases} \lambda e^{-\lambda x}, & x \ge 0, \\ 0, & x < 0. \end{cases} \]

CDF (cumulative distribution function)

\[ F(x) = P(X \le x) = \begin{cases} 1 - e^{-\lambda x}, & x \ge 0, \\ 0, & x < 0. \end{cases} \]

Survival function

\[ S(x) = P(X > x) = e^{-\lambda x}, \quad x \ge 0. \]

Mean, variance, and moments

Mean

\[ \mathbb{E}[X] = \frac{1}{\lambda} = \mu \]

Variance

\[ \mathrm{Var}(X) = \frac{1}{\lambda^2} \]

Standard deviation

\[ \sigma = \frac{1}{\lambda} \]

Tail probabilities and intervals

For any x ≥ 0:

\(P(X \le x) = 1 - e^{-\lambda x}\)
\(P(X > x) = e^{-\lambda x}\)

For an interval a ≤ X ≤ b with 0 ≤ a ≤ b:

\[ P(a \le X \le b) = F(b) - F(a) = \left(1 - e^{-\lambda b}\right) - \left(1 - e^{-\lambda a}\right) = e^{-\lambda a} - e^{-\lambda b}. \]

Quantiles and percentiles

The quantile function (inverse CDF) for 0 < p < 1 is:

\[ Q(p) = F^{-1}(p) = -\frac{1}{\lambda} \ln(1 - p). \]

Special cases:

Median (p = 0.5): \(Q(0.5) = \frac{\ln 2}{\lambda}\)
90th percentile (p = 0.9): \(Q(0.9) = -\frac{1}{\lambda}\ln(0.1)\)

Memoryless property

The exponential distribution is the only continuous distribution with the memoryless property:

\[ P(X > s + t \mid X > s) = P(X > t), \quad s, t \ge 0. \]

Proof sketch:

\[ P(X > s + t \mid X > s) = \frac{P(X > s + t)}{P(X > s)} = \frac{e^{-\lambda (s + t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(X > t). \]

Intuitively, if you have already waited s time units without an event, the additional waiting time until the next event still has the same exponential distribution as at the start.

Relationship with the Poisson distribution

If events occur according to a Poisson process with rate λ (the number of events in any interval of length t is Poisson with mean λt), then the waiting time X until the next event is exponential with rate λ:

\[ P(X > x) = P(\text{no events in } [0, x]) = e^{-\lambda x}. \]

Conversely, the sum of n independent exponential(λ) variables has a Gamma distribution with shape n and rate λ.

Exponential distribution FAQ

The standard inverse transform method works very well:

Generate U from Uniform(0, 1).
Set \(X = -\ln(1 - U) / \lambda\).

Then X has an exponential distribution with rate λ. Many software libraries implement this directly (e.g., rexp in R, numpy.random.exponential in Python).