Chapter 12: The Poisson distribution

adding on to mumps problem: suppose you saw 4 cases of Mumps in a given month. what are the chances of seeing 4 or more cases in any given month?

1 - ppois(q = 3, lambda = 0.1)

what is the Poisson distribution?

A Poisson distribution described the count X of occurrences of a defined event in fixed, finite intervals of time or space when: 1. Occurrences are all independent (that is, knowing that one event has occurred does not change the probability that another event may occur) 2. The probability of an occurrence is the same over all possible intervals of the same size

what do Poisson probabilities look like?

If X has the Poisson distribution with mean number of occurrences per interval µ, the possible values of X are 0, 1, 2, . . . .If k is any one of these values, then... P(X = k) = ((e^(-µ))(µ^(k))) / (k!)

example of Poisson distribution calculated by hand

In Iowa, the average monthly number of reported cases of mumps per year is 0.1. If we assume that cases of mumps are random and independent, the number X of monthly mumps cases in Iowa has approximately a Poisson distribution with µ = 0.1. The probability that in a given month there is no more than 1 mumps case is: P(X ≤ 1) = P(X = 0) + P(X = 1) so: ((e^(-0.1))(0.1^(0))) / (0!) + ((e^(-0.1))(0.1^(1))) / (1!) = 0.9048 + 0.0905 = 0.9953 Thus, we expect to only see 0 cases 90.5% of the months and 1 case 9.05% of the time

what is the main distinction bw Binomial and Poisson distributions?

Poisson random variables have no upper bound, whereas the upper bound of a binomial random variable X was n, the size of the sample.

example: polydactyly In the US, 1 in every 500 babies is born with an extra finger or toe. These events are random and independent. Suppose that the local hospital delivers an average of 268 babies per month. This means that for each month we expect to see 0.536 babies born with an extra finger or toe at that hospital (know how to arrive at 0.536). Let X be the count of babies born with an extra finger or toe in a month at that hospital.

X can take the value of 0, 1, 2 to infinity (in theory), so the X value is any positive integer X follows a Poisson distribution The mean of X is .536 and the standard deviation is the square root of the mean, which is .73

Polydactyly data in R

generate data across the 12 months of a year for this hospital: rpois(n = 12*5, lambda = 0.536)

recipe for Poisson distribution

goals: we want to find a probability, these are all fine: greater/less/equal/combo events occur independently the rate at which events occur is constant and the rate cannot be higher in some intervals and lower in other intervals

what is the mean and standard deviation of a Poisson random variable?

mean = µ variance = µ, so SD = µ^(1/2) if the mean is large, so is the SD, and this makes for a flat and wide probability distribution so if the mean is very large, the spread of the data will also be very large

what functions do you use to calculate Poisson distributions?

ppois() and dpois()

calculate the mumps example using R

ppois(q = 1, lambda = 0.1) lambda is the parameter or dpois(x = 0, lambda = 0.1) + dpois(x = 1, lambda = 0.1)

what is the Poisson distribution most commonly used for?

to model rare, random events or random events examined over small time intervals