Exponential Distribution

In Statistical and Probabilities, the exponential distribution is often used in order to model the latency before a specified event. For example, the exponential distribution could be used to describe the time passed between two telephone calls received at the office, or the time passed between two car accidents in which a given individual is implied.

Property of mental blank

An important property of the exponential distribution is the mental blank . This property results mathematically in the following equation:

$P (T > S + T \; |\; T > T) = P (T > S) \; \; \ forall \ S, T \ Ge 0.$

Let us imagine that X represents the lifespan of an electric bulb before it burns: the probability that it lasts at least s+t hours knowing that it already lasted T hours will be the same one as the probability of lasting S hours starting from its initial actuation. Into other words, the fact that it did not burn during T hours does not change anything with its life expectancy as from time T.

Specification of the exponential distribution

Density of probability

The Densité of probability of the exponential distribution takes the form

F (X; \ lambda) = \ left \ {\ begin {matrix} \ lambda e^ {- \ lambda X} &, \; X \ Ge 0, \ \ 0 &, \; X < 0. \end{matrix}\right.

where λ > 0 is a parameter. The distribution has as a support the interval

Function of distribution

The Fonction of distribution is given by

F (X; \ lambda) = \ left \ {\ begin {matrix} 1-e^ {- \ lambda X} &, \; X \ Ge 0, \ \ 0 &, \; X < 0. \end{matrix}\right.

Average and variance

The average or hope of X with parameter λ is

$\ mathbf {E} = \ frac {1} {\ lambda}$

its Variance is

$\ mathbf {V} = \ frac {1} {\ lambda^2}$ .

References

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