Poisson Distribution Lecture Notes PDF

# Lecture Seven ## Poisson Distribution * Recall that the Binomial distribution represents the probability of _K_ successes when a trial is represented _n_ times * The Poisson distribution is a special case where the number of trials _n_ approaches infinity, while the probability of success on eac...

# Lecture Seven ## Poisson Distribution * Recall that the Binomial distribution represents the probability of _K_ successes when a trial is represented _n_ times * The Poisson distribution is a special case where the number of trials _n_ approaches infinity, while the probability of success on each trial, _p_, is very small * A discrete random variable _X_ has a Poisson distribution with parameter _λ_ > 0 if its probability of mass function is given by: $P(K) = P(X = K) = \frac{\lambda^K e^{-\lambda}}{K!}$ for _K_ = 0, 1, 2, ... * _K_ represents the number of occurrences of an event * _K_! refers to the factorial of _K_ where * _K_! = _K_ x (_K_ - 1) x (_K_ - 2) x... x 2 x 1 * Therefore, the poisson distribution can be approximate the Binomial distribution when _n_ is large and _p_ is small * Like all the other probability distributions: * $\sum_{K=0}^\infty P(X = K) = 1$ * The expectation of a Poisson random variable _X_ is: * _E_(X) = _λ_ * The mean of a binomial variable is _np_ = _λ_ * Parameter _λ_ can be considered as the average number of occurrences in a given time period * The variance of _X_ is also _λ_: Var (_X_) = _λ_ ## When should we use Poisson Distribution? * When you want to model the number of times an (independent) event occurs within a fixed interval of time or space, and the events happen independently of each other. * e.g., Number of car crashes on a highway from 1-3pm * It is best suited when these events have a known average rate _λ_ but occur randomly. * e.g., Based on past data, there are 2 crashes on average from 1pm to 3pm * The poisson distribution is commonly used for count data. * The number of customer arrivals per hour * System failures per day * Number of emails received in a day ## Poisson Distribution Example * Let _X_ represent the number of wi-fi interruptions on your home network each day. * Suppose that interruptions occur at an average rate of 0.9 per day. * What is the probability with no more than two interruptions? * The count of flaws per day is modeled using a Poisson distribution with λ = 0.9 * With no more than two interruptions mean: _P_(_X_ ≤ 2) * _P_(_X_ ≤ 2) = _P_(_X_ = 0) + _P_(_X_ = 1) + _P_(_X_ = 2) * Plug in the λ into the poisson distribution for _X_ = 0, 1,2 * _P_(_X_ ≤ 2) = $\frac{0.9^0 e^{-0.9}}{0!} + \frac{0.9^1 e^{-0.9}}{1!} + \frac{0.9^2 e^{-0.9}}{2!}$ = 0.9372 * The 2!, 1!, and 0! will be provided on tests. ## Poisson Approximations * You can use the poisson approximation to the Binomial distributions when: * _n_ is large and _p_ is small * Here, _λ_ = _np_ * The poisson probability will approximately equal the binomial probability. * Suppose for the distribution _B_(_n_ = 1000, _p_ = 0.001). * Let _λ_ = 1000 x 0.001 = 1 * Approximate the Binomial distribution. * E.g., for _X_ = 5, using Binomial distribution, we have _P_(_X_ = 5) = 0.00307 * Let _X_ represent the number of defective hard drives produced at a factory. * Suppose that it is known that _P_ (defective hard drive) = 0.02. * Calculate the probability of _X_ > 3 using both Binomial distributions and Poisson approximation. * Here, _X_ has a binomial distribution with _n_ = 100 and _p_ = 0.02. * For having at least 3 successes out of 100 trials, it is easier to calculate using complement of _X_ < 3.