Refer to Exercise 3.122 . Assume that arrivals occur...

Refer to Exercise $3.122 .$ Assume that arrivals occur according to a Poisson process with an average of seven per hour. What is the probability that exactly two customers arrive in the twohour period of time between a. 2: 00 P.M. and 4: 00 P.M. (one continuous two-hour period)? b. 1: 00 P.M. and 2: 00 p.M. or between 3: 00 p.M. and 4: 00 P.M. (two separate one-hour periods that total two hours)?

Refer to Exercise $3.122 .$ Assume that arrivals occur according to a Poisson process with an average of seven per hour. What is the probability that exactly two customers arrive in the twohour period of time between
a. 2: 00 P.M. and 4: 00 P.M. (one continuous two-hour period)?
b. 1: 00 P.M. and 2: 00 p.M. or between 3: 00 p.M. and 4: 00 P.M. (two separate one-hour periods that total two hours)?

Key Concepts

Poisson Process

A Poisson process is a model that describes events occurring randomly over time or space, with the key property that events occur independently. In such a process, the probability of a given number of events occurring in a fixed interval of time depends only on the length of the interval and not on the starting point, making it especially useful for modeling arrivals or occurrences that are random in nature.

Poisson Distribution

The Poisson distribution is used to determine the probability of a specific number of events occurring within a fixed interval in a Poisson process. It is characterized by the parameter ? (lambda), which represents the average rate of events per unit time, and the probability mass function provides the likelihood of exactly k events taking place in the interval.

Scaling of the Parameter

Scaling the parameter ? is crucial when considering intervals of different lengths. When the base rate is given per unit time, the total expected number of events over a longer interval is obtained by multiplying the base rate by the length of that interval. For instance, if the base rate is seven per hour, the expected number over two hours is fourteen.

Independent Increments

The independent increments property of the Poisson process means that the numbers of events in non-overlapping intervals are statistically independent. This principle allows computations over separate time intervals, such as summing probabilities from disjoint intervals versus considering a continuous interval, to be handled independently.

Transcript

00:01 So we're assuming arrival times are based on a poisson process.

00:07 So since we are using poisson probability here, you are going to be applying the formula, the probability of x occurring equals the average raised to the x times e to the negative of the average all over x factorial.

00:29 So we want the probability of x being exactly two customers arriving in a two -hour period.

00:41 And we know that the average is seven per hour.

00:46 So if the average is seven in one hour, then it is 14 in two hours.

00:55 So since we are talking about a two -hour time period, our lambda is going to be, the two -hour time period would be 14.

01:08 So in part a, we are talking about a solid two -hour period.

01:14 So we want the probability that x equals two in one single two -hour time slot.

01:31 So in order to do that, we are going to apply our formula.

01:35 So we know that our lambda is going to be 14 in this instance.

01:40 So we have the probability that x is exactly 2 would equal 14 raised to the second power times e to the negative 14 power all over 2 factorial.

01:54 And when you calculate that out, you will get 8 .148980.

02:04 8147 times 10 to the negative fifth power.

02:10 So we could say the probability that x equals exactly two in a two -hour time period is going to be approximately 0 .000 -081 -5.

02:25 Now, in part b, we're going to change it up a little bit.

02:28 This time, we still want the probability of x equaling.

02:33 Two or exactly two, but this time it's in two separate one hour periods.

02:51 So this time our lambda is going to be seven because it's in a one hour time period instead of a two hour time period.

03:03 Now we've got to think of the different scenarios here.

03:06 So if we want the two separate time periods.

03:10 We could have it that there are two customers in the first hour and zero customers in the second hour, or we could have zero customers in the first hour and two customers in the second hour or we might have one customer in the first hour and one customer in the second hour.

04:18 So now that's going to change things a little bit...

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