0.13533528
0.86466472
0.13533528
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0.69314718
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0.69314718
0.13533528
0.86466472
0.13533528
1
1
0.69314718
1
1
0.69314718
The Exponential Distribution Calculator computes the probability density function (PDF), cumulative distribution function (CDF), survival function, and key statistics for the exponential distribution. The exponential distribution is the continuous probability distribution that models the time between events in a Poisson process, making it the continuous counterpart to the Poisson distribution.
If events occur at a constant average rate λ per unit time (following a Poisson process), then the waiting time between consecutive events follows an exponential distribution with rate parameter λ. This distribution appears everywhere in reliability engineering, queueing theory, telecommunications, physics, and survival analysis. It models the time until the next customer arrives, the lifespan of a light bulb, the time between radioactive decays, the duration of phone calls, and the time until the next earthquake.
The exponential distribution possesses the remarkable memoryless property: the probability of waiting an additional time t is independent of how long you have already waited. Mathematically, P(X > s + t | X > s) = P(X > t). This means that a component that has been running for 100 hours has the same probability of failing in the next hour as a brand-new component. This property is unique to the exponential distribution among continuous distributions and makes it the natural model for systems with constant failure rates.
The distribution is characterized by a single parameter λ > 0 (the rate), which equals the reciprocal of the mean. A higher λ means events occur more frequently and the expected waiting time 1/λ is shorter. The PDF decreases monotonically from λ at x = 0 toward zero as x increases, giving the characteristic right-skewed shape.
This calculator provides the PDF (density at x), CDF (probability of the event occurring by time x), survival function (probability of surviving beyond time x), plus the mean, variance, and standard deviation. Whether you are an engineer estimating component lifetimes, a manager analyzing service times, or a student learning probability theory, this tool delivers instant results for all exponential distribution computations.
The exponential distribution also plays a central role in Markov chain theory and continuous-time stochastic processes. In a continuous-time Markov chain, the time spent in each state before transitioning is exponentially distributed, with different states potentially having different rate parameters. In network reliability analysis, exponential failure times allow closed-form computation of system reliability for series, parallel, and complex configurations. The hazard rate (instantaneous failure rate) of the exponential distribution is constant at λ, which is why it models the useful life period of electronic components (between burn-in and wear-out phases in the bathtub curve). For practitioners in operations research, the M/M/1 queue model assumes exponential interarrival and service times, yielding elegant formulas for average queue length, waiting time, and server utilization.
The exponential distribution PDF for x ≥ 0 is:
$$f(x) = \lambda e^{-\lambda x}$$
The CDF (probability that X is at most x) is:
$$F(x) = P(X \leq x) = 1 - e^{-\lambda x}$$
The survival function (complementary CDF) is:
$$S(x) = P(X > x) = e^{-\lambda x}$$
The key statistics are:
$$E[X] = \frac{1}{\lambda}, \quad \text{Var}(X) = \frac{1}{\lambda^2}, \quad \sigma = \frac{1}{\lambda}$$
Note that the mean and standard deviation are equal (both 1/λ), which is a distinctive property of the exponential distribution. The median is ln(2)/λ ≈ 0.693/λ, which is always less than the mean due to the right-skewed shape.
The PDF f(x) gives the instantaneous rate of probability at point x (density, not probability). The CDF P(X ≤ x) is the probability that the event occurs within time x. For example, if CDF = 0.865, there is an 86.5% chance the event happens by time x. The survival function P(X > x) is the probability of the event NOT occurring by time x, useful in reliability analysis (probability a component is still working at time x). The mean 1/λ is the average waiting time between events.
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Results
A light bulb with failure rate λ = 0.001 per hour has a mean lifespan of 1000 hours. After 500 hours, there is a 39.3% chance it has failed and a 60.7% chance it is still working.
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If customers arrive at rate 3 per minute, the average time between arrivals is 0.333 minutes (20 seconds). The probability of waiting more than 1 minute between customers is only 4.98%.
The memoryless property states that P(X > s + t | X > s) = P(X > t). In plain language: if you have already waited s units without an event, the probability of waiting an additional t units is the same as starting fresh. The exponential is the only continuous distribution with this property. It implies a constant failure rate (no aging or wear-out), which is realistic for electronic components but not for mechanical parts.
The exponential distribution can be parameterized by rate λ (events per unit time) or by mean β = 1/λ (average time between events). Both are equivalent: f(x) = λe^(-λx) = (1/β)e^(-x/β). Our calculator uses the rate parameterization. If you know the mean waiting time, enter λ = 1/mean.
They are two sides of the same coin. If events follow a Poisson process with rate λ (count of events in an interval is Poisson(λt)), then the time between events follows an Exponential(λ) distribution. The Poisson models the count; the exponential models the waiting time.
The exponential is inappropriate when the failure rate is not constant. If components wear out over time (increasing failure rate) or have initial burn-in failures (decreasing failure rate), use the Weibull distribution instead. The Weibull generalizes the exponential: when its shape parameter equals 1, it reduces to the exponential.
The survival function S(x) = P(X > x) = e^(-λx) gives the probability of surviving (or waiting) beyond time x. In reliability engineering, S(1000) gives the probability a component lasts beyond 1000 hours. In medicine, it gives the probability a patient survives beyond time x. It is the foundation of survival analysis and Kaplan-Meier estimation.
The p-th percentile (quantile) is found by inverting the CDF: x_p = -ln(1-p)/λ. For example, the median (50th percentile) is x₀.₅ = ln(2)/λ ≈ 0.693/λ. The 90th percentile is x₀.₉ = ln(10)/λ ≈ 2.303/λ. The 95th percentile is -ln(0.05)/λ ≈ 2.996/λ.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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