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Poisson distribution - Exercise set 1

This exercise set contains some solved exercises on the Poisson distribution. The theory needed to solve these exercises is introduced in the lecture entitled Poisson distribution.

Exercise 1.1

The time elapsed between the arrival of a customer at a shop and the arrival of the next customer has an exponential distribution with expected value equal to 15 minutes. Furthermore, it is independent of previous arrivals. What is the probability that more than 6 customers arrive at the shop during the next hour?

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If a random variable has an exponential distribution with parameter $lambda $, then its expected value is equal to $1/lambda $. Here[eq1]Therefore, $lambda =4$. If inter-arrival times are independent exponential random variables with parameter $lambda $, then the number of arrivals during a unit of time has a Poisson distribution with parameter $lambda $. Thus, the number of customers that will arrive at the shop during the next hour (denote it by X) is a Poisson random variable with parameter $lambda =4$. The probability that more than 6 customers arrive at the shop during the next hour is:[eq2]and the value of [eq3] can be calculated with a computer algorithm, for example with the MATLAB command:[eq4]

Exercise 1.2

At a call center, the time elapsed between the arrival of a phone call and the arrival of the next phone call has an exponential distribution with expected value equal to 15 seconds. Furthermore, it is independent of previous arrivals. What is the probability that less than 50 phone calls arrive during the next 15 minutes?

nav_button Solution

If a random variable has an exponential distribution with parameter $lambda $, then its expected value is equal to $1/lambda $. Here[eq5]where, in the last equality, we have taken 15 minutes as the unit of time. Therefore, $\lambda =60$. If inter-arrival times are independent exponential random variables with parameter $lambda $, then the number of arrivals during a unit of time has a Poisson distribution with parameter $lambda $. Thus, the number of phone calls that will arrive during the next 15 minutes (denote it by X) is a Poisson random variable with parameter $\lambda =60$. The probability that less than 50 phone calls arrive during the next 15 minutes is:[eq6]and the value of [eq7] can be calculated with a computer algorithm, for example with the MATLAB command:[eq8]

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