ExponentialDistribution - Maple Help

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:      for   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:    for

Properties

 PDF ${\mathrm{\lambda }}^{}{ⅇ}^{-\mathrm{λt}}$ The probability density function CDF The cumulative distribution function Mean E(X) The expected value of a random variable Variance Var(X) $\frac{1}{{\mathrm{λ}}^{2}}$ Represented by the symbol ${\mathrm{σ}}^{2}$, representing how much variation or spread exists from the mean value

where $\mathrm{\lambda }$ = 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) =  = 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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