0.36787944
0.63212056
0.36787944
1
1
1
1
—
0.693147
0
0.36787944
0.63212056
0.36787944
1
1
1
1
—
0.693147
0
The Exponential Distribution Calculator computes probability density, cumulative probability, and key statistics for the exponential distribution. This continuous probability distribution models the time between independent events that occur at a constant average rate, making it fundamental in reliability engineering, queuing theory, and survival analysis.
The exponential distribution is parameterized by a single rate parameter $$\lambda > 0$$. Its probability density function (PDF) is defined as:
$$f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0$$
The cumulative distribution function (CDF) gives the probability that the random variable $$X$$ takes a value less than or equal to $$x$$:
$$F(x; \lambda) = 1 - e^{-\lambda x}, \quad x \geq 0$$
The exponential distribution is the only continuous distribution with the memoryless property: $$P(X > s + t \mid X > s) = P(X > t)$$. This means that the probability of waiting an additional time $$t$$ is independent of how long you have already waited. This property makes the exponential distribution the continuous analogue of the geometric distribution.
The mean (expected value) of the exponential distribution is $$E[X] = \frac{1}{\lambda}$$, and the variance is $$\text{Var}(X) = \frac{1}{\lambda^2}$$. The median is $$\frac{\ln 2}{\lambda} \approx \frac{0.6931}{\lambda}$$, which is always less than the mean since the distribution is right-skewed.
The exponential distribution is closely related to the Poisson process: if events follow a Poisson process with rate $$\lambda$$, then the waiting time between consecutive events follows an exponential distribution with the same rate. It is also a special case of the gamma distribution with shape parameter $$k = 1$$, and a special case of the Weibull distribution with shape $$k = 1$$.
In reliability engineering, the exponential distribution models the lifetime of components with a constant failure rate. In queuing theory, it describes inter-arrival and service times. In physics, radioactive decay times follow an exponential distribution. In finance, it models the time between trades or default events.
Enter the rate parameter $$\lambda$$ and a value $$x \geq 0$$. The calculator computes the PDF, CDF, survival function, mean, variance, standard deviation, and median of the distribution.
The calculator evaluates the exponential PDF as $$f(x) = \lambda e^{-\lambda x}$$ and the CDF as $$F(x) = 1 - e^{-\lambda x}$$ for the given rate parameter and value. The survival function is $$S(x) = 1 - F(x) = e^{-\lambda x}$$. Statistical measures are derived directly from $$\lambda$$.
The PDF value indicates the relative likelihood of observing exactly $$x$$. The CDF gives the probability that the observed value is at most $$x$$. The survival function gives the probability of exceeding $$x$$. A higher $$\lambda$$ concentrates the distribution closer to zero, producing a steeper PDF curve and faster CDF approach to 1.
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Results
With λ = 2 (average 2 events per unit time), the probability of observing X ≤ 0.5 is about 63.2%. The mean waiting time is 0.5 time units.
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Results
With a failure rate of 0.1 per hour, the mean lifetime is 10 hours. The probability of surviving beyond 10 hours is about 36.8%.
The rate parameter $$\lambda$$ represents the average number of events per unit time (or per unit of whatever is being measured). A higher $$\lambda$$ means events happen more frequently, and the expected waiting time $$1/\lambda$$ is shorter.
The memoryless property means $$P(X > s + t \mid X > s) = P(X > t)$$. Given that you have already waited $$s$$ time units, the probability of waiting an additional $$t$$ units is the same as the original probability of waiting $$t$$ units from the start.
If the number of events in a fixed interval follows a Poisson distribution with rate $$\lambda$$, then the time between consecutive events follows an exponential distribution with the same rate $$\lambda$$. They describe the same process from different perspectives.
Use it when modeling waiting times between independent events occurring at a roughly constant rate. Common applications include time between customer arrivals, time to component failure (with constant hazard rate), and time between radioactive decays.
The exponential distribution is right-skewed, meaning it has a long tail to the right. This pulls the mean above the median. Specifically, the median is $$\frac{\ln 2}{\lambda} \approx 0.693/\lambda$$ while the mean is $$1/\lambda$$.
No. The exponential distribution assumes a constant hazard (failure) rate. For increasing or decreasing failure rates, use the Weibull distribution, which generalizes the exponential with an additional shape parameter.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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