Question

You have developed a self-service kiosk capable of serving about 15 clients per hour. You have been told that the average rate of customers using this kiosk is about 10 customers per hour. You also know that the number of customers who approach the kiosk per hour follows the Poisson distribution.

1. Write out the pmf of the Poisson RV in this case and solve for 20 customers approaching the kiosk.

2. Use an R function to find a probability for the above.

3. Have R generate random numbers following the above distribution for 100,000 intervals. What is the maximum number of customers approaching the kiosk in your simulation?

Answer 1

Solution:

Question 1

We are given that number of customers approach the kiosk per hour follows Poisson distribution.

λ = 10

P(X=x) = λ^x*exp(-λ)/x!

P(X=x) = 10^x*exp(-10)/x!

Now, we have to find P(X=20)

P(X=20) = 10^20*exp(-10)/20!

P(X=20) = 10^20* 0.0000453999297624849/20!

P(X=20) = 0.001866081

Question 2

We have to find P(X=20) by using R.

We are given λ = 10

R-command and output is given as below:

> dpois(20,10,log = FALSE)

[1] 0.001866081

Required probability = 0.001866081

Question 3

Required R command and maximum number of customers is given as below:

> max(rpois(100000,10))

[1] 28

The maximum number of customers approaching the kiosk in simulation by using R is given as 28.