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Stat 260 - Lecture 12 Recap of Last Class • We introduced the binomial distribution and binomial random variables. Today: • Begin with an example illustrating the use of the binomial distribution. • We will introduce another important distribution known as the Poisson distribution. • Also introduce the Poisson process. • Go through some examples... The Poisson Distribution • A discrete random variable X is said to have a Poisson distribution with parameter λ > 0 if the pmf of X is e−λλx P (X = x) = p(x; λ) = , x = 0, 1, 2, 3, ... x! • A Poisson random variable takes values on the nonnegative integers. • The distribution is used often in statistics for modelling count data. Simple Example: If X ∼ Poisson(λ = 2) then what is P (X > 0)? P (X > 0) = 1 − P (X ≤ 0) = 1 − P (X = 0) e−220 = 1 − p(0; 2) = 1 − ≈ 0.865 0! Relationship to the binomial • The Poisson distribution is often used as an approximation to the binomial distribution. • Proposition: Suppose X ∼ Bin(n, p) so that the pmf is given by b(x; n; p). If we let n → ∞ and p → 0 in such a way that np approaches a value λ > 0. Then b(x; n, p) → p(x; λ). • In other words, if X ∼ Bin(n, p) with n large and p small, we can approximate the pmf of X with the Poisson pmf p(x; λ = np). • Rule of thumb to use this approximation: n ≥ 100, p ≤ 0.01 and np ≤ 20. Expected Value and Variance - Poisson • If X has a Poisson distribution with parameter λ, then E[x] = λ and V [X] = λ. • Note that for a Poisson random variable E[x] = V [X]. The Poisson Process • Counting Process: There are many situations in which we observe the occurrence of events of a particular type over time. For example: 1. In a soccer game, the events of interest might be goals scored, which occur throughout the course of the game. 2. In a hospital emergency room, the events of interest might be new hospital admissions that occur throughout the course of the day. 3. At an airport, the events of interest might be the arrival of airplanes during some period of time. In all these cases, we observe some process which yields recurring events: goals, hospital admissions or airplane arrivals over time and we observe the time at which each event occurs. • Often we are not interested in the exact times at which events occur; instead, we are interested in the total number of events during some time interval of length t. 1. Soccer example: # goals in the 1st minute of the 2nd half. 2. Hospital example: # of new admissions between 5:30 PM and 6:00PM. 3. Airport example: # of airplanes arriving during any 20 minute interval during the day. • In this case, we are interested in a random process, N (t), which counts the number of events in any interval of time having length t. • We call N (t) a counting process. For any fixed t0, the quantity N (t0) is a random variable. • For example starting at t = 0, the counting process N (t) might count the total number of points scored by Kobe Bryant after t minutes of a lakers game. So here, N (12), N (24), N (36) and N (48) represent the total number of points scored after each quarter - each of these is a random variable. • Poisson Process: A counting process, N (t), is called a Poisson process with rate parameter α if all of the following three conditions hold: 1. The number of events in nonoverlapping intervals occur independently of one another. 2. The probability distribution of the number of events in any interval of time depends only on the length of the interval. 3. The probability of x events in an interval of length t is P (N (t) = x) = (αt) −αt e x x! that is, for each t we have N (t) ∼ Poisson(λ = αt) Examples... Summary of Today’s Class • We introduced the Poisson distribution and Poisson process. Homework: • Problem set 12