2.6 Applications of Poisson distribution
A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time, if these events happen at a known average rate and independently of the time since the last event
The Poisson distribution may be used to approximate the Binomial, if the probability of success is “small” (less than or equal to \(0.01\)) and the number of trials is “large” (greater than or equal to \(50\))
Other rules of thumb are also suggested by different authors, but all recognize that the Poisson distribution is the limiting distribution of the Binomial as \(n\) increases and \(p\) approaches zero
The formula for computing probabilities that are from a Poisson process is
\[\begin{align} P(X=x_i)&=\frac{\lambda^{x_i}e^{\lambda}}{x!}\\ \\ x_i&=\text{the number of successes } (x_i:~0,~1,~2,...) \\ \\ \lambda&=\text{the average number of events that occur per time interval} \\ \\ e&=\text{constant equel to } 2.71828 \\ \\ x_i!&=\text{the porduct of first x positive integers (factorial)} \\ \tag{2.11} \end{align}\]
- If a random variable \(X\) follows a Poisson distribution we use notation
\[\begin{equation} X \sim P(\lambda) \end{equation}\]
- The expected value of a Poisson distribution is
\[\begin{equation} E(X)=\lambda=np \end{equation}\]
- The variance of a Poisson distribution is
\[\begin{equation} Var(X)=E(X)=\lambda \end{equation}\]
- To compute Poisson probabilities in Excel you can use function
=POISSON.DIST(x;lambda;FALSE)with setting the cumulative distribution function to FALSE (last argument of the function)
Example 2.7 On average, a clothing store gets \(120\) customers per day
- What is the probability of getting \(150\) customers in one day?
\(P(X=150)=\)_______
- What is the probability of getting \(35\) customers in the first four hours? Assume the store is open \(12\) hours each day.
\(P(X=35)=\)_______
Example 2.8 The chance of an IRS audit for a tax return is about \(2\%\) per year. Suppose that \(100\) people are randomly chosen.
- How many people are expected to be audited?
\(E(X)=\)_______
- Find the probability that no one was audited.
\(P(X=0)=\)_______
- Find the probability that at least two were audited.
\(P(X \geq 2)=1-P(X=0)-P(X=1)=\)_______
Example 2.9 Calculate the probability that there will be \(220\) infected people by COVID-19 in a one day using the Excel function =POISSON.DIST() if a total of \(2800\) new cases are recorded in the last \(14\) days.