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The most important applications of the exponential distribution are situations where the Poisson process is applied. the Poisson process allows us to use the discrete distribution called the Poisson distribution. Remember that the Poisson distribution is used to calculate the probability of specific numbers of "events" during a period or space private cio. In many applications the random variable is time or quantity of space. The relationship between the exponential distribution (often called the negative exponential) and the Poisson process is very simple. The Poisson distribution will be developed as a single- parameter distribution with parameter A, where A will be interpreted as the mean number of events per unit of "time". Now consider the random variable described by the time required for it to precede the first event. If we use the Poisson distribution, we see that the probability that some event does not occur, in the period up to time 1, is given by

The most important applications of the exponential distribution are situations where the Poisson process is applied. the Poisson process allows us to use the discrete distribution called the Poisson distribution. Remember that the Poisson distribution is used to calculate the probability of specific numbers of "events" during a period or space private cio. In many applications the random variable is time or quantity of space. The relationship between the exponential distribution (often called the negative exponential) and the Poisson process is very simple. The Poisson distribution will be developed as a single- parameter distribution with parameter A, where A will be interpreted as the mean number of events per unit of "time". Now consider the random variable described by the time required for it to precede the first event. If we use the Poisson distribution, we see that the probability that some event does not occur, in the period up to time 1, is given by

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Hello! I need help explaining with general words the following fragment of a text about Poisson and mention what is the relationship between the Poisson and exponential distribution according to the text

I put it in two photos so that it can be appreciated better
 
 
 
| The most important applications of the exponential distribution are situations where the Poisson process is applied. the Poisson process allows us to use the discrete distribution called the Poisson distribution. Remember that the Poisson distribution is used to calculate the probability of specific numbers of "events" during a period or space private cio. In many applications the random variable is time or quantity of space. The relationship between the exponential distribution (often called the negative exponential) and the Poisson process is very simple. The Poisson distribution will be developed as a single- parameter distribution with parameter A, where A will be interpreted as the mean number of events per unit of "time". Now consider the random variable described by the time required for it to precede the first event. If we use the Poisson distribution, we see that the probability that some event does not occur, in the period up to time 1, is given by P(0; λt) = eat (at)⁰ 0! = e-At 1
Transcribed Image Text:| The most important applications of the exponential distribution are situations where the Poisson process is applied. the Poisson process allows us to use the discrete distribution called the Poisson distribution. Remember that the Poisson distribution is used to calculate the probability of specific numbers of "events" during a period or space private cio. In many applications the random variable is time or quantity of space. The relationship between the exponential distribution (often called the negative exponential) and the Poisson process is very simple. The Poisson distribution will be developed as a single- parameter distribution with parameter A, where A will be interpreted as the mean number of events per unit of "time". Now consider the random variable described by the time required for it to precede the first event. If we use the Poisson distribution, we see that the probability that some event does not occur, in the period up to time 1, is given by P(0; λt) = eat (at)⁰ 0! = e-At 1
2 We can now use the above and let X be the time to the first Poisson event. The probability that the length of time until the first event exceeds x is the same as the probability that no Poisson event occurs at x. The latter, of course, given by e-λt. As a result, P (X> x) = e-^t Thus, the cumulative distribution function for X is given by: P (0 ≤ x ≤ x) = 1 - e-^t Now to recognize the presence of the exponential distribution, we can differentiate the previous cumulative distribution function to obtain the density function f(x) = λe-^x Being the exponential distribution density function with λ = 1/B
Transcribed Image Text:2 We can now use the above and let X be the time to the first Poisson event. The probability that the length of time until the first event exceeds x is the same as the probability that no Poisson event occurs at x. The latter, of course, given by e-λt. As a result, P (X> x) = e-^t Thus, the cumulative distribution function for X is given by: P (0 ≤ x ≤ x) = 1 - e-^t Now to recognize the presence of the exponential distribution, we can differentiate the previous cumulative distribution function to obtain the density function f(x) = λe-^x Being the exponential distribution density function with λ = 1/B
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