5. Customers arrive at coffee shop according to a Poisson process at a mean rate of 5 customers every 10 minutes. (a) Le

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5 Customers Arrive At Coffee Shop According To A Poisson Process At A Mean Rate Of 5 Customers Every 10 Minutes A Le 1 (129.5 KiB) Viewed 32 times

5. Customers arrive at coffee shop according to a Poisson process at a mean rate of 5 customers every 10 minutes. (a) Let X denote the waiting time (in minutes, counted from 7 am) until the first customer arrives. What is the pdf of X? Name the distribution if it has a name. (b) Find the probability that the owner of the coffee shop will have to wait longer than 5 minutes before the arrival of the first customer. (C) Find the conditional probability that the owner of the coffee shop will have to wait longer than 5 minutes before the arrival of the first customer, given that no customer came in the first three minutes. (d) What is the probability that the owner of the coffee shop will have to wait longer than 28.41 minutes before the arrival of the 10th customer? Use two different approaches to solve this question by using the relationship be- tween the gamma, Poisson, and chi-square distributions. You are allowed to use R to obtain the numerical answers.