The arrival pattern of order messages of the interactive...

The arrival pattern of order messages of the interactive order entry system of the Dizzy Disc record company has a Poisson distribution with an average of 25 arrivals per minute during the peak period. What is the probability that more than 30 orders will arrive in one minute of the peak period? Use the normal approximation if you don't have facilities such as APL, MINITAB, or EXPLORE to make the exact calculation.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability model that describes the number of events occurring in a fixed interval of time or space, under the assumptions that these events occur with a known constant mean rate and independently of the time since the last event. It is widely used in various fields to model count-based occurrence data.

Normal Approximation

The normal approximation is a technique used to approximate a discrete distribution, like the Poisson, with a continuous normal distribution when the mean is sufficiently large. This approximation simplifies the computation of probabilities, especially when calculation tools are limited.

Continuity Correction

When using the normal approximation to estimate probabilities for a discrete distribution, a continuity correction is applied to adjust for the fact that the normal distribution is continuous. This involves modifying the discrete variable by ±0.5 in order to more accurately approximate the probability of the equivalent discrete event.

Transcript

00:01 In this question, we are given a poisson process with a rate of three per hour.

00:10 And in the first part of the question, what is the probability of having 10 events occur in the interval from 0 to 5 hours, given that we've had four events in the time interval from 0 to 2 hours? so the numbers of events that are occurring in these processes are poisson random variables, and we know that because it's been described as a poisson process.

00:37 Remember that for a poisson random variable, the probability of so many events occurring within a given time interval is equal to the following.

01:02 So for part a, we are looking for the probability that the number of events occurring within five hours is equal to 10, given that the number of events that occur in the first two hours is four.

01:23 So the time intervals that we're talking about are from 0 to 2 hours and from 0 to 5 hours.

01:39 And as you can see, these are overlapping time intervals.

01:42 The first time interval is a subset of the second one.

01:47 So from our probability laws, we can say that this is equal to the probability of the number of events occurring between 0 and 5 hours, equaling 10, an intersection with the number of events occurring between zero and two hours, equalling four, divided by the probability that the number of events occurring in the first two hours is equal to four.

02:23 Now, another way to look at this is if we are given that four events have occurred between zero and two hours, then for 10 events to have occurred by five hours, we need an additional six events to occur between the times of two hours and five hours.

02:47 So we can write this as the probability of the number of events occurring between five and two hours is equal to six.

03:19 So now we're dealing with two intervals.

03:22 One is from zero to two, and the second is from two to five.

03:31 And these are non -overlapping intervals.

03:34 And for a personal process, the number of events that occur in non -overlapping intervals is independent, which means that we can write the intersection of the two events in the numerator as the product of two probabilities.

03:51 So we can say that the probability of the number of events occurring in three hours being equal to six times the probability that the number of events that occurs in two hours is equal to four.

04:14 And then divided by the probability that the number of events that occurs in two hours is equal to four.

04:24 And obviously this gets a lot nicer now that we can factor these out.

04:34 So now we can just solve for the probability of having six events in three hours, given a process rate of three per hour.

04:45 So this is e to the negative lambda t, which comes out to negative 9.

04:50 So lambda is 3 and t is 3 times lambda t to the exponent 6 over 6 factorial...

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