# Exponential Distribution Calculator

The exponential distribution probability density that predicts waiting times, failure rates, and other events in which the rate of occurrence remains constant over time. For instance, the time it takes for a call to be answered at a call center may be an exponentially distributed random variable. The exponential distribution is used in many branches of science and telecommunications. The equations of the probability density function and cumulative distribution function are

pdf(x) = ce^{-cx} [0, ∞)

CDF(x) = 1 - e^{-cx} [0, ∞)

where *c* is a positive constant, the rate parameter.

Using the formulas below, you can compute statistics of the exponential distribution (mean, median, variance, and standard deviation) and the probability that an exponentially distributed random variable X is between two numbers X_{1} and X_{2}.

#### Computing P(X_{1} < X < X_{2})

If X is an exponentially distributed random variable, you compute P(X_{1}< X < X

_{2}) by plugging X

_{1}and X

_{2}into the CDF, then subtracting the two values.

P(X

_{1}< X < X

_{2}) = e

^{-cX1}- e

^{-cX2}.

The exponential distribution has the property of memorylessness, meaning that

P(A+B < X | A < X) = P(B < X).

In other words, the probability that X is greater than a certain threshold (A+B), given that it is higher than another lower threshold (A), is equal to the probability that X is greater than the difference between the thresholds (A+B-A = B).

In practical terms, this means the probability of an event occurring is the same at any point in time, regardless of how long you have been waiting prior.

#### Computing the Mean

To calculate the mean of a continuous probability density function p(x), you evaluate the integral ∫xp(x) dx over its domain. In the exponential distribution, the domain is [0, ∞) and the mean μ isμ = 1/c.

#### Computing the Variance and Standard Deviation

The variance of a continuous probability distribution is found by computing the integral ∫(x-μ)²p(x) dx over its domain. For the exponential distribution, the variance σ² is given byσ² = 1/c².

The standard deviation σ is simply the positive square root of the variance, so σ = 1/c. In the exponential distribution, the mean and standard deviation are equal.

#### Computing the Median

The median of a continuous distribution function is a number*m*such that the integral ∫

^{m}

_{o}p(x) dx = 1/2. For the exponential distribution the median is given by the equation

m = (LN(2))/c.

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