The Poisson Distribution

1. Definition of Poisson Random Variable

Def: A Poisson random variable with mean A and probability mass function: P(X=i) = E^{-A}frac{A^i}{i!}

  • mean: A
  • variance: A

2. Binomial Approximate Poisson

Let X_i be an indicator variable of either 0 or 1. And assume Pr(X_i = 1) = A/N for all i. Then E[X] = A.
Based on the assumptions, we have a binomial distribution:
P(sum^N_{i=1} X_i = k) = binom{N}{k}(A/N)^k(1-A/N)^{N-k}
                                            = frac{N!}{(N-k)!k!} (A/N)^k (1-A/N)^{N-k}
                                            = e^{-A} A^k/k! (A Poisson random variable)

3. Example

P(X_i =1) =1/365.
X = sum^{400}_{i=1} X_i is approximately Poisson with mean 400/365
Then P(X geq 2) = 1 - P(X<2) = 1-e^{-A} - Ae^{-A}

4. Priliminary Definitions

Def : A stochastic process is a collection of random variable (RV) indexed by time {X(t), t in T}.

  • If T is continuous set, the process is a continuous time stochastic process (e.g., Poisson Process)
  • If T is countable, then the process is a discrete time stochastic process (e.g., Markov chain)
Def. A counting process is a stochastic process {N(t); t geq 0} such that
  • N(t) in {0,1,cdots, 2} (that is, N(t) is non-negative integer)
  • If s < t, the N(s) leq N(t) ( that is, N(t) is non-decreasing in t)
  • For s<t, N(t) - N(s) is the number of events occurring in the time interval (s,t].
Def: A counting process has stationary increments if the distribution of the number of events in an interval depends on the length of the interval, but not on the starting point of the interval. That is, P(N(s+t) - N(s) =  n) does not depend on s. Intuitively, the interval can be “slide” around without changing its stochastic nature.

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