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 = λⅇ−λt 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=PX≤t=1−ⅇ−λ t for t > 0
Properties
PDF
λⅇ−λt
The probability density function
CDF
1−ⅇ−λ t
The cumulative distribution function
Mean E(X)
1λ
The expected value of a random variable
Variance Var(X)
1λ2
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.
Example
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−λt ⅆt = 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) =
More MathApps
MathApps/ProbabilityAndStatistics
Download Help Document
