Exponential Distribution:

The exponential distribution is a continuous memory less distribution that describes the time between events in a Poisson process. The continuous analogue of the geometric distribution gives exponential distribution. The exponential distribution models time between successive events over a continuous time interval, whereas the Poisson distribution deals with events that happen over a fixed period of time.

The exponential distribution is mostly used for testing product reliability which deals with the amount of time a product lasts. For example, the amount of time (beginning now) until an earthquake occurs, the length( in minutes) of long distance business telephone calls, and the amount of time(in months) a car battery lasts.

The events occur in disjoint intervals (non-overlapping)

Two or more events cannot occur simultaneously

Each event occurs at a constant rate

A continuous random variable X is said to have an exponential distribution with parameter λ>0, if its PDF is given by

f_X(x) =  

Where, e = the natural number e, λ = mean time between events, x = a random variable.

The formula for the cumulative distribution function of the exponential distribution is:
F(x) = 1 − e ^{− λx}    where x ≥ 0; λ > 0

Mean =

Variance =

Suppose you are testing new software, and a bug causes errors randomly at a constant rate of three times per hour. The probability that the first bug will occur within the first ten minutes is?

Let constant rate or intensity be λ = 3 / hour and t = 1/6 hours (10 minutes)

P(X < 1/6) =  = 0.393

The probability that the first bug will occur in the next 10 minutes is 0.393.

Let us see this example in Excel using the function, EXPONDIST(x,lambda,cumulative)

X   – Value of the function in question, Lambda   – constant value,

Cumulative =