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The Exponential Distribution

Main Concept

The exponential distribution is a continuous memoryless distribution that describes the time between events in a Poisson process. It is a continuous analogue of the geometric distribution.


In order for an event to be described by the exponential distribution, there are three conditions in which the event must hold:


Independence: The events occur in disjoint intervals (non-overlapping)


Individuality: Two or more events cannot occur simultaneously


Uniformity: Each event occurs at a constant rate

If random variable X follows an exponential distribution, the distribution of waiting times between events is defined by the following probability density function:


ft = λⅇλt   for t  > 0


Where: l is the constant rate or intensity at which the event occurs at and t is the length of time between two events.

The cumulative distribution function is defined as:


ft=PXt=1ⅇλ t for t  > 0





The probability density function


1ⅇλ t

The cumulative distribution function

Mean E(X)


The expected value of a random variable

Variance Var(X)


Represented by the symbol σ2, representing how much variation or spread exists from the mean value

where λ = is the intensity or the rate at which an event occurs.



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


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

P(X < 1/6) = 016λe&lambda;t &DifferentialD;t = 0.393

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



Change the intensity of the event l and time t to observe the change in the exponential distribution and the corresponding probability value:

rate of event (l) =  

time between events (t) =



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