Lesson 7: Poisson distribution

TB sections 3.4

Meike Niederhausen and Nicky Wakim

2025-10-27

Learning Objectives

1. Calculate probabilities for different events using a Poisson distribution

Where are we?

Introduction to the Poisson distribution

• The Poisson distribution is often used to model count data (# of successes), especially for rare events

  • It is a discrete distribution!

 

• It is used most often in settings where events happen at a rate \(\lambda\) per unit of population and per unit time

• Example: historical records of hospitalizations in New York City indicate that an average of 4.4 people are hospitalized each day for an acute myocardial infarction (AMI)

  • We can plot the distribution of hospitalizations on each day

 

Poisson distribution

• Suppose events occur over time in such a way that

  1. The probability an event occurs in an interval is proportional to the length of the interval.

  2. Events occur independently at a rate \(\lambda\) per unit of time.

• Then the probability of exactly \(x\) events in one unit of time is \[ P(X = k) = \frac{e^{-\lambda}\lambda^{k}}{k!}, \,\, k = 0, 1, 2, \ldots \]

• For the Poisson distribution modeling the number of events in one unit of time:

  • The mean is \(\lambda\).

  • The standard deviation is \(\sqrt{\lambda}\).

• Shorthand for a random variable, \(X\), that has a Poisson distribution: \[X \sim \text{Pois}(\lambda)\]

Poisson distribution: R commands

R commands with their input and output:

R code	What does it return?
rpois()	returns sample of random variables with specified Poisson distribution
dpois()	returns value of probability density at certain point of the Poisson distribution
ppois()	returns cumulative probability of getting certain point (or less) of the Poisson distribution
qpois()	returns number of cases corresponding | to desired quantile

Example: probabilities from a Poisson distribution (1/4)

Typhoid fever

Suppose there are on average 5 deaths per year from typhoid fever over a 1-year period.

1. What is the probability of 3 deaths in a year?

2. What is the probability of 2 deaths in 0.5 years?

3. What is the probability of more than 12 deaths in 2 years?

Example: probabilities from a Poisson distribution (2/4)

Typhoid fever

Suppose there are on average 5 deaths per year from typhoid fever over a 1-year period.

1. What is the probability of 3 deaths in a year?

• \(\lambda = 5\) and we want \(P(X = 3)\)

\[P(X=3) = \frac{e^{-5}5^{3}}{3!} = 0.1404\]

dpois(x = 3, lambda = 5)

[1] 0.1403739

Example: probabilities from a Poisson distribution (3/4)

Typhoid fever

Suppose there are on average 5 deaths per year from typhoid fever over a 1-year period.

2. What is the probability of 2 deaths in 0.5 years?

\(\lambda = ?\) and we want \(P(X = 2)\)

• \(\lambda=5\) was the rate for one year. When we want the rate for half year, we need to calculate a new \(\lambda\):

  • \(\lambda = \dfrac{5 \text{ deaths}}{1 \text{ year}}\cdot\dfrac{1 \text{ year}}{2 \text{ half-years}}=\dfrac{2.5 \text{ deaths}}{1 \text{ half-year}}\)

\[P(X=2) = \frac{e^{-2.5}2.5^{2}}{2!} = 0.0.2565\]

dpois(x = 2, lambda = 2.5)

[1] 0.2565156

Example: probabilities from a Poisson distribution (4/4)

Typhoid fever

Suppose there are on average 5 deaths per year from typhoid fever over a 1-year period.

3. What is the probability of more than 12 deaths in 2 years?

\(\lambda = ?\) and we want \(P(X > 12)\)

• Need to calculate a new \(\lambda\) for 2 years: \(\lambda = \dfrac{5 \text{ deaths}}{1 \text{ year}}\cdot\dfrac{2 \text{ years}}{1 \text{ two-years}}=\dfrac{10 \text{ deaths}}{1 \text{ two-year}}=\dfrac{10 \text{ deaths}}{2 \text{ years}}\)

\[P(X>12) = 1 - P(X \leq 12) = 1 - \sum_{k=0}^{12}\frac{e^{-10}10^{k}}{k!} = 0.2084\]

1 - ppois(q = 12, lambda = 10)

[1] 0.2084435

ppois(q = 12, lambda = 10, 
      lower.tail = F)

[1] 0.2084435

Poisson approximation of binomial distribution

• Poisson distribution can be used to approximate binomial distribution when \(n\) is large and \(p\) is small
  • When Normal approximation does not work
• Binomial distributions for different \(n\) (columns) and \(p\) (rows)