Exponential Distribution Calculator
The exponential distribution is the 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. You encounter it in reliability engineering when modelling how long a component works before failing, in queuing theory when modelling customer arrival times, in telecommunications when modelling call durations, and in nuclear physics when modelling radioactive decay intervals. The key parameter is lambda, the rate at which events occur. If a server handles on average 0.5 customers per minute, lambda = 0.5 and the mean wait time is 1/0.5 = 2 minutes. The probability that the waiting time X is less than some value t is given by the cumulative distribution function: P(X less than or equal to t) = 1 minus e to the power negative lambda t. The survival function P(X greater than t) = e to the power negative lambda t gives the probability of waiting longer than t. A key feature of the exponential distribution is the memoryless property: the probability of waiting an additional time t is the same regardless of how long you have already waited. This calculator takes the rate parameter lambda and a time value t, and returns the CDF, survival probability, probability density at t, mean, median, variance and standard deviation.
lambda must be positive. t must be non-negative. Mean = 1/lambda, Median = ln(2)/lambda, Variance = 1/lambda squared.
How it works
For an exponential distribution with rate parameter λ and time t ≥ 0:
PDF: f(t) = λ e-λt
CDF: P(X ≤ t) = 1 - e-λt
Survival: P(X > t) = e-λt
Mean = 1/λ
Median = ln(2)/λ ≈ 0.6931/λ
Variance = 1/λ²
Std dev = 1/λ
Note that for the exponential distribution, mean = standard deviation = 1/λ. The coefficient of variation is always exactly 1.
Worked example
With lambda = 0.5 and t = 3: CDF = 1 - e-0.5 × 3 = 1 - e-1.5 = 1 - 0.2231 = 0.7769. P(X > 3) = e-1.5 = 0.2231. PDF at t = 0.5 × e-1.5 = 0.1116. Mean = 1/0.5 = 2.0000. Median = ln(2)/0.5 = 1.3863. Variance = 1/0.25 = 4.0000. Std dev = 1/0.5 = 2.0000.
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