Relationship between Exponential and Poisson distribution 지수분포와 포아송 분포의 관계

Exponential Distribution

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When X follows the exponential distribution, X~EXP(λ)

random variable X expresses the time until the next event occurs.

So exponential distribution is continuous distribution because it deals with time which is continuous.

 

 

PDF(Probability Density Function) of Exponential Distribution

 

Expected value and Variance of the exponential distribution

 

 

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Poisson Distribution

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When Y has the poisson distribution, Y~POI(λ)

random variable Y expresses the number of event occurrences in a fixed time period.

So poisson distribution is discrete distribution because random variables should be discrete numbers.

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PMF(Probability Mass Function) of Poisson Distribution.

 

Expected value and Variance of Poisson distribution

 

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Relationship between Exponential & Poisson distribution

 

Both distributions need the same parameter 'lambda' λ which denotes the average rate of events during the specific period that is, the expected value of poisson distribution with parameter λ.

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Relationship between exponential distribution and Poisson distribution can be explained by definining random variable of exponential distribution as 'time gap between poisson distribution with λ.'

Let me give you an example with known parameter lambda (3/10)

proof of relationship 

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CDF(cumulative density function) of Exponential Distribution.

 

When X~EXP(λ) and Y~POI(λ)

 

P(X>t) : time interval between events is more than 't' period.

It needs more than 't' period to have one event to occur.

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P(Y=0): 0 events during 't' period.

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you should know that P(X>t) and P(Y=0) have same meaning.

and you can prove that there is a relationship between exponential and poisson distribution using the fact.