19  Poisson Distributions

• Binomial and Negative Binomial models have several restrictive assumptions that might not be satisfied in practice
• Poisson models are more flexible and are often used to model the distribution of random variables that count the number of “relatively rare” events that occur over a certain interval of time/in a certain location

Example 19.1 Recall the matching problem with a general \(n\): there are \(n\) objects that are shuffled and placed uniformly at random in \(n\) spots with one object per spot. Note: \(n\) is a fixed and known number, but we are considering different values of it.

We have seen that \(\text{E}(X)=1\) for any \(n\) and that unless \(n\) is small (e.g., \(n \le 6\)) the approximate distribution of \(X\) is the Poisson(1) distribution.

Table 19.1: Poisson(1) distribution, the approximate distribution of \(X\), the number of matches in the matching problem for general \(n\) and uniformly random placement of objects in spots.

x	P(X=x)
0	0.3679
1	0.3679
2	0.1839
3	0.0613
4	0.0153
5	0.0031
6	0.0005
7	0.0001

1. Starting with \(x=0\), describe the distribution of \(X\) in terms of relative likelihoods:
   • 1 is [blank] times as likely as 0
   • 2 is [blank] times as likely as 1
   • 3 is [blank] times as likely as 2
   • 4 is [blank] times as likely as 3
   • 5 is [blank] times as likely as 4
   • 6 is [blank] times as likely as 5
   • In general, \(x\) is [blank] times as likely as \(x-1\), for \(x=1, 2, 3, \ldots\)
     
     
2. Specify the pmf of \(X\).
   
   
   
   
   
3. Use the pmf to compute \(\text{E}(X)\).
   
   
   
   
   
4. Use the pmf to compute \(\text{Var}(X)\).
   
   
   
   
   

• A random variable \(X\) has a Poisson distribution with parameter \(\mu>0\) if the possible values are \(0, 1, 2, \ldots\) and the pmf follows the \[ \text{Poisson(}\mu\text{) pattern:}\quad x \text{ is } \frac{\mu}{x} \text{ as likely as } x-1 \quad \text{ (for } x = 1, 2, 3, \ldots) \]
• As a function of \(x=0, 1, 2, \ldots\), a Poisson(\(\mu\)) pmf increases for \(x<\mu\) and then decreases for \(x >\mu\)
  • If \(\mu\) is a whole number, then the Poisson pmf assigns the same probability to the values \(\mu\) and \(\mu-1\).
  • If \(\mu<1\) then \(x=0\) has the highest probability and the pmf decreases from there.
• A discrete random variable \(X\) has a Poisson distribution with parameter \(\mu>0\) if and only if its probability mass function \(p_X\) satisfies \[ p_X(x) = \frac{e^{-\mu}\mu^x}{x!}, \quad x=0,1,2,\ldots \]
  • The Poisson pattern implies that the shape of a Poisson pmf as a function of \(x\) is given by \(\mu^x/x!\). \[ p(x) = p(0)\frac{\mu^x}{x!}, \qquad x = 0, 1, 2, \ldots \]
  • The constant \(p(0) = e^{-\mu}\) simply renormalizes the pmf so that the probabilities sum¹ to 1.
• If \(X\) has a Poisson(\(\mu\)) distribution then \[\begin{align*} \text{E}(X) & = \mu\\ \text{Var}(X) & = \mu \end{align*}\]

Example 19.2 Let \(X\) be the number of home runs hit (in total by both teams) in a randomly selected Major League Baseball game. Assume that \(X\) has a Poisson(2.4) distribution.

1. Interpret the parameter 2.4 in context.
   
   
   
   
   
2. Compute \(\text{P}(X = 2.4)\).
   
   
   
   
   
3. Identify the most likely value of \(X\).
   
   
   
   
   
4. Specify the pmf of \(X\).
   
   
   
   
   
5. Compute \(\text{P}(X = 0)\) and interpret the value in context.
   
   
   
   
   
6. Construct a table and spinner corresponding to the distribution of \(X\).
   
   
   
   
   

Example 19.3 Suppose that the number of fatal crashes in SLO County in a week follows, approximately, a Poisson(0.53) distribution, independently from week to week. Let \(X\) be the number of fatal crashes in a period of 4 consecutive weeks (close enough to a “month”).

1. What is \(\text{E}(X)\)? You should be able to compute it without knowing the distribution of \(X\).
   
   
   
   
   
2. Compute \(\text{P}(X = 0)\).
   
   
   
   
   
3. Compute \(\text{P}(X = 1)\).
   
   
   
   
   
4. How could you use simulation to approximate the distribution of \(X\)?
   
   
   
   
   
5. Use simulation to approximate the distribution of \(X\). Does the approximate distribution follow (approximately) the Poisson pattern?
   
   
   
   
   
6. Use a Poisson pmf to compute \(\text{P}(X = 5)\).
   
   
   
   
   

• Poisson aggregation. If \(X\) and \(Y\) are independent, \(X\) has a Poisson(\(\mu_X\)) distribution, and \(Y\) has a Poisson(\(\mu_Y\)) distribution, then \(X+Y\) has a Poisson(\(\mu_X+\mu_Y\)) distribution.
• If component counts are independent and each has a Poisson distribution, then the total count also has a Poisson distribution.

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1. Recall the series expansion \(e^{\mu} = \sum_{x=0}^\infty \frac{\mu^x}{x!}\)