The Poisson random variable arises naturally when we have to count the number of events occurring over a specific time interval. We see that this kind of distribution is intimately related to exponential random variables, which are dealt with in Section 7.6.3, and with the Poisson stochastic process, introduced in Section 7.9. For now, the best way to understand the Poisson distribution is by thinking about the random process of customer arrivals to a service facility. The main feature of such a process is the average arrival rate, i.e., the expected number of customers requesting service per unit time. Let us denote this rate by λ. If the arrival rate does not change in time and t is the length of a time interval, the expected number of customers arriving during that interval is λt.
Let X be the number of customers arriving during an interval of unit length. Its support is {0,1, 2, 3,…}, and its expected value is the arrival rate λ. The exact distribution of X depends on many things, including the random time elapsing between two consecutive arrivals. If interarrival times satisfy certain sensible properties, then the number of customers arriving during any time interval of unit length has Poisson distribution. The PMF of a Poisson random variable with parameter λ is
Fig. 6.6 The PMF of a Poisson random variable with parameter λ = 5.
Figure 6.6 shows the PMF of a Poisson random variable with parameter λ = 5. Recalling Taylor’s expansion of the exponential function22, it is easy to see that Poisson probabilities add up to 1:
Calculating the expected value is also fairly straightforward:
It can also be shown that Var(X) = λ.
Example 6.14 A customer service center receives, on average, λ = 3 calls per hour. Using the PMF of a Poisson distribution, we may calculate the probability of receiving an arbitrary number of calls within one hour. Since e−3 = 0.0498, we have:
The probability of receiving two or more calls
At present, we cannot fully illustrate the conditions under which the number of events occurring with given rate can be modeled by a Poisson variable. They are linked to the memoryless property of the exponential random variable, which we defer to Section 8.5.2. Nevertheless, at an intuitive level, we may say that a Poisson distribution arises when
- The arrival process is stationary, i.e., the arrival rate does not change over time
- The arrival process is purely random, i.e., there is no regular pattern like the one arising from deterministic and regular arrivals
- The arrival numbers in two disjoint time intervals are two independent random variables
There is another way to shed some light on the Poisson distribution, which emphasizes its link with the binomial random variable. Consider a binomial random variable with parameters p → 0 and n → +∞. In other words, we have a huge number of Bernoulli trials, but the probability of success for each one is tiny. Then, it can be shown that the distribution of such a Bernoulli random variable tends to the distribution of a Poisson random variable with parameter λ = pn.
Problems
6.1 Consider a generalization of the Bernoulli random variable, i.e., a variable taking values x1 with probability p and x2 with probability 1 − p. Which values of p maximize and minimize variance?
6.2 Using Eq. (6.15), prove that the expected value of a geometric random variable X with parameter p is E[X] = 1/p. (Hint: Use the result in Example 2.40.)
6.3 Using the binomial expansion formula (2.3), prove that the PMF of the binomial distribution [see Eq. (6.16)] adds up to one.
6.4 You are about to launch a new product on the market. If it is a success you will make $16 million; otherwise, you lose $5 million. The probability of success is 65%. You could increase chances of success by delaying product launch in order to improve product design; this would take 6 months and would cost an additional $1 million. In order to account for the delay and the time value of money, we should discount cash flows at a rate of 3% (the rate refers to the 6-month period and is applied to profit/loss). What is the minimal improvement in success probability that makes the delay worthwhile?
6.5 According to an accurate survey, 40% of people checked at the exit of a well-known pub have made excessive use of alcoholic drinks. If we take a random sample of 25 persons, what is the probability that at least 4 of them are flagged?
6.6 Batteries produced by a company are known to be defective with a probability of 0.02. The company sells batteries in packages of eight and offers a money-back guarantee that at most one of them is defective. What is the probability that a package is returned? If a customer buys three packages, what is the probability that exactly one of them will be returned?
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