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I Proof to the Expression of Poisson Distribution

  1. Oct 7, 2016 #1
    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
     
  2. jcsd
  3. Oct 7, 2016 #2

    andrewkirk

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    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.
     
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