## - Exponential Distribution

A classical situation in which an exponential distribution arises is as follows: Consider a Poisson process with mean λ where we count the events occurring in a given interval of time or space. Let X denote the waiting time until the first event to occur. Then, for a given x > 0,

P (X > x) = P (no event in (0, x))

= exp(−xλ), and hence

P (X ≤ x) = 1− exp(−xλ). (15.1) The above distribution is the exponential distribution with mean waiting time b = 1/λ. The probability density function (pdf) is given by

f(x|b) = 1 b exp

( −x b

) , x > 0, b > 0. (15.2)

Suppose that the waiting time is known to exceed a threshold value a, then the pdf is given by

f(x|a, b) = 1 b exp

( −x− a

b

) , x > a, b > 0. (15.3)

The distribution with the above pdf is called the two-parameter exponential distribution, and we referred to it as exponential(a, b). The cumulative distribution function is given by

F (x|a, b) = 1− exp(−(x− a)/b), x > a, b > 0. (15.4)

Mean: b+a Median a− ln(.5)b

Variance: b2 Mode: a

Coefficient of Variation: b b+a

Coefficient of Skewness: 2

Coefficient of Kurtosis: 9

Moment Generating Function: (1− bt)−1, t < 1 b when a = 0

Moments about the Origin: E(Xk) = bkΓ(k + 1) = bkk!, k = 1, 2, . . .; a = 0

The dialog box [StatCalc→Continuous→Exponential→Probabilities...] computes the tail probabilities, percentiles, moments, and other parameters of an exponential distribution.