Mathematics, Statistics & Geometry

Exponential Distribution Calculator

Calculate the probability density (PDF) and cumulative probability (CDF) for continuous exponential decay and wait-time distributions.

Cumulative Prob P(X ≤ x)
0.632
Probability Density (PDF)0.184
Expected Mean (E[X])2
Variance4
Calculation StepsRate (λ) = 0.5, x = 2 Cumulative Distribution Function (CDF): P(X ≤ x) = 1 - e^(-λx) CDF = 1 - e^(-0.5 * 2) = 1 - e^(-1) = 0.632121 Probability Density Function (PDF): f(x) = λe^(-λx) PDF = 0.5 * e^(-0.5 * 2) = 0.183940 Mean = 1 / λ = 1 / 0.5 = 2.0000

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Modeling Wait Times and Decay

The Exponential Distribution Calculator evaluates continuous probability scenarios involving time and distance. Whether you are modeling radioactive decay or the wait time between customer arrivals, this tool provides the precise PDF and CDF probabilities.

f(x)=λe−λxandF(x)=1−e−λx\begin{aligned} f(x) = \lambda e^{-\lambda x} \quad \text{and} \quad F(x) = 1 - e^{-\lambda x} \end{aligned}

Where:
λ\lambda=
The average number of events per unit interval
x=
The specific wait time or distance being measured
e=
The mathematical constant (~2.718)

The Poisson Connection

The Exponential distribution is deeply linked to the Poisson distribution.

  • If the number of events per hour follows a Poisson distribution with rate λ\lambda.
  • Then the time between those events follows an Exponential distribution with the same rate λ\lambda.

Real-World Applications

  • Reliability Engineering: Modeling the 'Mean Time Between Failures' (MTBF) for electronic components or mechanical parts.
  • Queueing Theory: Predicting how long a customer will wait in line at a bank, or how long a network packet will wait in a router buffer.
  • Physics: Calculating the half-life and decay probability of unstable radioactive isotopes.

Frequently Asked Questions

It is a continuous probability distribution used to model the 'time between events' in a Poisson process. It is famous for its 'memoryless' property.

It means the probability of an event occurring in the next minute is exactly the same regardless of how long you have already been waiting. A lightbulb doesn't 'remember' that it is old.

The PDF f(x) gives the relative likelihood of an exact specific wait time. The CDF F(x) gives the probability that the wait time will be LESS THAN or EQUAL TO x.

They are exact inverses. If a server receives 5 requests per minute (rate = 5), the expected wait time between requests is 1/5th of a minute (0.2 minutes).

No, it is heavily right-skewed. The highest probability density is always at x=0, and it decays exponentially as time goes on.

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