## 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

$$$X \sim P(\lambda)$$$

• The expected value of a Poisson distribution is

$$$E(X)=\lambda=np$$$

• The variance of a Poisson distribution is

$$$Var(X)=E(X)=\lambda$$$

• 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

1. What is the probability of getting $$150$$ customers in one day?

$$P(X=150)=$$_______

1. 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.

1. How many people are expected to be audited?

$$E(X)=$$_______

1. Find the probability that no one was audited.

$$P(X=0)=$$_______

1. 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.

$$P(X=220)=POISSON.DIST(220;200;FALSE)=0.0102$$