Question

Customers make purchases at a convenience store, on average, every fourteen minutes. It is fair to assume that the time between customer purchases is exponentially distributed. Jack operates the cash register at this store.
a-1. What is the rate parameter
λ

? (Round your answer to 4 decimal places.)

Rate parameter
λ

a-2. What is the standard deviation of this distribution? (Round your answer to 1 decimal place.)

Standard deviation

b. Jack wants to take a ten-minute break. He believes that if he goes right after he has serviced a customer, he will lower the probability of someone showing up during his ten-minute break. Is he right in this belief?

Yes No

c. What is the probability that a customer will show up in less than ten minutes? (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)

Probability

d. What is the probability that nobody shows up for over forty minutes? (Round intermediate calculations to 4 decimal places and final answer to 4 decimal places.)

Probability

Answer 1

Answer:

The answers to the questions are;

a-1. The rate parameter, λ is 0.0714 .

a-2. The standard deviation is 14.

b. No

c. The probability that a customer will show up in less than ten minutes is 0.5104.

d. The probability that nobody shows up for over forty minutes is 0.0574.

Step-by-step explanation:

a-1. The exponential distribution probability density function is given by

PDF = $\lambda e^{-\lambda x}$ and the cumulative distribution function is given by

CDF = $1- e^{-\lambda x}$

Therefore the rate parameter the rate at which customers make purchases, that is the number of customers per minute

Therefore, the number of customers per minute = rate parameter = 1/14

The rate parameter, λ = 0.0714 .

a-2. In exponential distribution, which is the probability of the amount of time  between two events where the rate of occurrence of events is constant, known as a Poisson point process, we have;

The mean = 1/λ = standard deviation

Therefore, the standard deviation is 14.

b. No

c. The probability that a customer will show up in less than ten minutes is given by the cumulative distribution function as follows

P (10<X) =  $1- e^{-\frac{10}{14} }$ = 0.5104.

d. The probability that nobody shows up for over forty minutes is given by the area on the right of the exponential distribution function curve.

Therefore, we have

P (X>40) =  $1- (1- e^{-\frac{40}{14} })$ = 0.0574

The probability that nobody shows up for over forty minutes is 0.0574.