Exponential distribution: Difference between revisions
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imported>Michael Hardy (→A basic introduction to the concept: This example is dubious. Cleaned up notation.) |
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The '''[[exponential distribution|exponential distribution]]''' is any member of a class of [[continuous probability distribution|continuous probability distributions]] assigning probability | The '''[[exponential distribution|exponential distribution]]''' is any member of a class of [[continuous probability distribution|continuous probability distributions]] assigning probability | ||
: <math>e^{-x/\mu} \,</math> | : <math>e^{-x/\mu} \,</math> | ||
to the interval <nowiki>[</nowiki>''x'', ∞<nowiki>)</nowiki>. | to the interval <nowiki>[</nowiki>''x'', ∞<nowiki>)</nowiki>, for ''x'' ≥ 0. | ||
It is well suited to model lifetimes of things that don't "wear out", among other things. | It is well suited to model lifetimes of things that don't "wear out", among other things. | ||
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==A basic introduction to the concept== | ==A basic introduction to the concept== | ||
The main and unique characteristic of the exponential distribution is that the [[conditional probability|conditional probabilities]] | The main and unique characteristic of the exponential distribution is that the [[conditional probability|conditional probabilities]] satisfy P(''X'' > ''x'' + ''s'' | ''X'' > ''x'') = P(''X'' > ''s'') for all ''x'', ''s'' ≥ 0. | ||
===Formal definition=== | ===Formal definition=== | ||
Let ''X'' be a real, positive stochastic variable with [[probability density function]] | Let ''X'' be a real, positive stochastic variable with [[probability density function]] | ||
: <math>f(x)= \lambda e^{-\lambda x} \ | : <math>f(x)= \lambda e^{-\lambda x}\,</math> | ||
Then ''X'' follows the exponential distribution with parameter <math>\lambda</math>. | for ''x'' ≥ 0. Then ''X'' follows the exponential distribution with parameter <math>\lambda</math>. | ||
==See also== | ==See also== | ||
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*[[Poisson distribution]] | *[[Poisson distribution]] | ||
== | ==References== | ||
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Latest revision as of 16:00, 14 August 2024
The exponential distribution is any member of a class of continuous probability distributions assigning probability
to the interval [x, ∞), for x ≥ 0.
It is well suited to model lifetimes of things that don't "wear out", among other things.
The exponential distribution is one of the most important elementary distributions.
A basic introduction to the concept
The main and unique characteristic of the exponential distribution is that the conditional probabilities satisfy P(X > x + s | X > x) = P(X > s) for all x, s ≥ 0.
Formal definition
Let X be a real, positive stochastic variable with probability density function
for x ≥ 0. Then X follows the exponential distribution with parameter .
