Question

4. Customers enter a store at a rate of 3 customers per hour.

a) Compute the probability that at least two, but no more than five customers enter the store in a given hour.

b) Compute the probability that it takes more than 30 minutes for the first customer to enter the store this hour.

c) Suppose you know that exactly 1 customer entered the store during a given hour. Compute the probability that the customer entered the store between minute 10 and minute 30.

d) Let Xk be the number of customers that enter a store during hour k. Suppose you recorded how many customers entered the store each hour for the last 60 hours. What is the approximate distribution of X¯? Make sure to specify the parameter(s) of the distribution.

Answer 1

a)

Rate = 3 customers per hour

X ~ Poisson(3)

Probability that at least two, but no more than five customers enter the store in a given hour

= P(2 X 5)

= P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

= 0.2240418 + 0.2240418 + 0.1680314 + 0.1008188

= 0.7169338

b)

For 30 minutes, rate = 3 * 0.5 = 1.5

Y ~ Poisson(1.5)

Probability that it takes more than 30 minutes for the first customer to enter the store this hour

= Probability that no customer to enter the store in 30 minutes

= P(Y = 0) =

= 0.2231302

c)

Let Z be number of customers arrive between minute 10 and minute 30.

Time duration between minute 10 and minute 30 is 20 minutes = 1/3 hour

Rate = 3 * 1/3 = 1

Z ~ Poisson(1)

Let W be number of customers arrive between minute 0 and 10 and minute 30 to minute 60.

Time duration between between minute 0 and 10 and minute 30 to minute 60 is 40 minutes = 2/3 hour

Rate = 3 * 2/3 = 2

W ~ Poisson(2)

Probability that the customer entered the store between minute 10 and minute 30 given that exactly 1 customer entered the store during a given hour

= Probability that the customer entered the store between minute 10 and minute 30 AND exactly 1 customer entered the store during a given hour / Probability that exactly 1 customer entered the store during a given hour

= Probability that the customer entered the store between minute 10 and minute 30 and no customers entered the store between minute 0 and 10 and minute 30 to minute 60 / Probability that exactly 1 customer entered the store during a given hour

= P(Z = 1) * P(W = 0) / P(X = 1)

= * /

= 1/3

d)

Since the sample size is greater than 30, the approximate distribution of X¯ will be Normal distribution.

The parameters of the distribution are,

Mean

Variance