Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016
Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What is your gender? (1 for male 0 for female) Tossing a coin (1 for Head 0 for Tail) Sickness or health.
Example Y: smoking status (Y=1: smoker, Y=0: non-smoker) 29% of the adults in the U.S. are smokers, so the probability distribution of Y:
Example Y: smoking status (Y=1: smoker, Y=0: non-smoker) 29% of the adults in the U.S. are smokers, so the probability distribution of Y: Randomly selected two individuals, X: number of current smokers in this pair. Probability distribution of X:
Example Y: smoking status (Y=1: smoker, Y=0: non-smoker) 29% of the adults in the U.S. are smokers, so the probability distribution of Y: Randomly selected two individuals, X: number of current smokers in this pair. Probability distribution of X: Randomly selected three individuals, X: number of current smokers among these. Probability distribution of X:
Binomial Distribution If we have a sequence of n independent outcomes of the Bernoulli random variable Y, each with probability of success p, then the total number of successes X is a binomial random variable. The fixed numbers n and p are parameters of this distribution.
Binomial Distribution If we have a sequence of n independent outcomes of the Bernoulli random variable Y, each with probability of success p, then the total number of successes X is a binomial random variable. The fixed numbers n and p are parameters of this distribution. n and p for the previous example.
Binomial Distribution The binomial distribution satisfies the following assumptions: There a fixed number of trials, n, each of which results in one of the two mutually exclusive and exhaustive outcomes: success or failure. The outcomes of n trials are independent, so that the outcome on any particular trial does not influence the outcome on any other trial. The probability of success is constant from trial to trial; we denote this probability by p.
Binomial Distribution When X B(n, p), we use the following formula to find the P(X = x), ( ) n p x (1 p) n x, x = 0, 1,..., n P(X = x) = x 0, otherwise
P(X = 0), P(X = 1), P(X = 2), and P(X = 3) for the smokers example.
Expected Value and Variance For X B(n, p), E(X ) = np V (X ) = np(1 p) σ X =
Poisson Distribution The Poisson distribution is used to model discrete events that occur infrequently in time or space, hence, it is sometimes called the distribution of rare events. Let X be a random variable that represents the number of occurrences of some event of interest over a given interval. Since X is a count, it can theoretically assume any integer value between 0 and infinity. X has a Poisson distribution with parameter λ > 0 and the pmf of X is { e λ λ x, x = 0, 1,... P(X = x) = x! 0, otherwise λ denotes the average number of occurrences of the event in an interval. It is denoted as X Poisson(λ).
Example X: number of people in a population of 10,000 who will be involved in a motor vehicle accident each year. Probability of a particular individual involved is 0.00024. P(X=0), P(X=2), and P(X 7).
When a Poisson r.v is appropriate: As a limit of Binomial. As a count: number of typos per page in a book. number of people in a population of 10,000 who will be involved in a motor vehicle accident each year.
Limit of Binomial Let X B(n, p) with n 50 np 5 then X is said to follow Poisson with λ = np, that is X Poisson(np).
Using the Poisson to approximate the Binomial Given that 5% of a population are left-handed, use the Poisson distribution to estimate the probability that a random sample of 100 people contains 2 or more left-handed people. X = No. of left handed people in a sample of 100 X Bin(100, 0.05) Poisson approximation X Poi(λ) with λ = 100 0.05 = 5
Assumptions of Poisson Distribution The poisson distribution satisfies the following assumptions: The probability that a single event occurs within an interval is proportional length of the interval. Within a single interval, an infinite number of occurrence of the event are theoretically possible. We are not restricted to a fixed number of trials. The events occur independently both within the same interval and between consecutive intervals.
Example Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability of observing 4 births in a given hour at the hospital? Let X = No. of births in a given hour
Example cont d (Changing the size of the interval) Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability that we observe 5 births in a given 2 hour interval? Y = No. of births in a 2 hour period
Expected Value and Variance For X Poisson(λ), E(X ) = λ V (X ) = λ σ X =