Proof to the Expression of Poisson Distribution

In summary, a Poisson distribution is used to describe the number of times an event occurs in a given interval, with the assumption that events occur independently and at a constant rate. The probability of an event in an interval is proportional to the length of the interval. This distribution can be deduced as a limiting case of the binomial distribution, where the limit as the number of events approaches infinity results in the Poisson CDF formula.
  • #1
Anakin Skywalker
1
0
Hello.
Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements:
  • X is the number of times an event occurs in an interval and X can take values 0, 1, 2, …
  • The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
  • The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
  • Two events cannot occur at exactly the same instant.
  • The probability of an event in an interval is proportional to the length of the interval.
And in this case the probability related to x is given by the expression below:
poisson-formula.png


I would like to know how this expression is deduced.

P.S.: I used the informations in the wikipedia's page, so I'm not so sure that these topics are right.
https://en.wikipedia.org/wiki/Poiss..._Poisson_distribution_an_appropriate_model.3F
 
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  • #2
The Poisson distribution can be deduced as a limiting case of the binomial distribution.

Let the probability of exactly one event occurring in a period of length ##\delta t## be ##g(\delta t)##, and define
$$\lambda\triangleq \lim_{\delta t\to 0}\frac{g(\delta t)}{\delta t}$$

Then if ##X(T)## is the number of events occurring in a period of length ##T## and ##n## is a positive integer, we have
$$Pr(X(T)\leq x)=Pr (B_n\leq x)$$
where ##B_n## is a binomial random variable with parameters ##(n,g(T/n))##.

Hence ##Pr (B_n\leq x)## is constant over ##n## and hence its limit as ##n\to\infty## exists and is equal to ##Pr(X(T)\leq x)##.

But the number of events occurring in a period of length ##T## is also equal to the Poisson random variable ##Y(\lambda,T)## for the number of events in time period ##T## with frequency ##\lambda##.

So we have
$$Pr(Y(\lambda,T)\leq x)=\lim_{n\to\infty}Pr(B_n\leq x)$$
and it can be shown that the expression on the RHS is equal to the formula for the Poisson CDF.

I've left some steps out because I'm a bit rushed this morning, but this should at least give an idea of how it's done.
 

1. What is the Poisson distribution?

The Poisson distribution is a statistical probability distribution that is used to model the number of events that occur in a fixed time interval when the events are independent and occur at a constant rate.

2. How is the Poisson distribution expressed mathematically?

The Poisson distribution is expressed as P(x) = (e^(-λ) * λ^x) / x!, where x is the number of events, e is the base of the natural logarithm, and λ is the rate parameter.

3. What is the role of proof in understanding the expression of Poisson distribution?

Proof is essential in understanding the expression of Poisson distribution as it provides mathematical evidence and reasoning behind the formula. It helps to show how the formula is derived and how it accurately models the probability of events occurring.

4. What are the assumptions made when using the Poisson distribution?

The assumptions made when using the Poisson distribution include that the events occur at a constant rate, the events are independent, and the probability of an event occurring in a small time interval is proportional to the length of the interval.

5. In what real-life situations is the Poisson distribution commonly used?

The Poisson distribution is commonly used in situations where events occur randomly and independently, such as in the number of phone calls received in a call center, the number of customers entering a store, or the number of accidents on a highway in a given time interval.

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