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Orders arrive at a Web site according to a Poisson process
Chapter 3, Problem 172E(choose chapter or problem)
Problem 172E
Orders arrive at a Web site according to a Poisson process with a mean of 12 per hour. Determine the following:
(a) Probability of no orders in five minutes.
(b) Probability of 3 or more orders in five minutes.
(c) Length of a time interval such that the probability of no orders in an interval of this length is 0.001.
Questions & Answers
QUESTION:
Problem 172E
Orders arrive at a Web site according to a Poisson process with a mean of 12 per hour. Determine the following:
(a) Probability of no orders in five minutes.
(b) Probability of 3 or more orders in five minutes.
(c) Length of a time interval such that the probability of no orders in an interval of this length is 0.001.
ANSWER:
Solution :
Step 1 of 3:
Given for an interval of the length of time T with the parameter .
We know that from the mean number of orders arrive at a website per hour are 12.
So .
Then the poisson variable X’s probability function is
P(X=x) =
Our goal is:
a). We need to find the probability of no orders in 5 minutes.
b). We need to find the probability of 3 or more in 5 minutes.
c). We need to find the probability of no orders in an interval of this length is 0.001.
a). Now we have to find the probability of no orders in 5 minutes.
We know that .
Then,
1
Probability of getting no orders, so P(X=0).
P(X=0) =
P(X=0) =
P(X=0) = 0.367
Therefore, the probability of no orders in 5 minutes is 0.3678.