Suppose entries to an order-entry system of the Shootemup...

Suppose entries to an order-entry system of the Shootemup Arms Company arrive at the central processor with a Poisson pattern at an average rate of 30 per minute. (a) Given that an order entry transaction has just arrived, what is the average time until the fourth succeeding transaction arrives? (b) What is the probability that it will take longer than 10 seconds for this entry to arrive? Less than 5 seconds? (c) Will the answers to the above questions change if the point in time at which measurement begins is 1 second after a transaction arrives?

Suppose entries to an order-entry system of the Shootemup Arms Company arrive at the central processor with a Poisson pattern at an average rate of 30 per minute.
(a) Given that an order entry transaction has just arrived, what is the average time until the fourth succeeding transaction arrives?
(b) What is the probability that it will take longer than 10 seconds for this entry to arrive? Less than 5 seconds?
(c) Will the answers to the above questions change if the point in time at which measurement begins is 1 second after a transaction arrives?

Key Concepts

Poisson Process

A Poisson process is a statistical model used to describe events that occur independently and randomly in time at a constant average rate. It is characterized by having events that occur with a constant mean rate and whose counts in non-overlapping intervals are independent.

Exponential Distribution

The exponential distribution represents the interarrival times in a Poisson process, meaning it models the time between successive events. Its memoryless property implies that the likelihood of an event occurring in the future does not depend on how much time has already elapsed.

Gamma Distribution

The gamma distribution generalizes the exponential distribution for the waiting time until the nth event in a Poisson process. It is obtained as the sum of several independent exponential random variables, with its shape parameter corresponding to the number of events and its scale parameter related to the rate of occurrence.

Memoryless Property

The memoryless property means that the probability of an event occurring in the future is independent of any past events or elapsed time. This characteristic, inherent to the exponential distribution, ensures that the statistical behavior of future waiting times remains the same regardless of when measurement starts.

Transcript

00:01 Orders arrive at a website according to a pusan process.

00:06 So because we have this pusan process, we are doing a pouzsson probability.

00:13 And you will be utilizing the formula, the probability of a certain count that x equals a certain value, is equivalent to e to the negative lambda times lambda to the x.

00:34 X power all over x factorial.

00:39 Now in here we know that there is a mean of 12 orders in one hour.

00:54 So we've got to find our lambda based on the amount of time.

00:59 So in part a, we are talking about a five minute time period.

01:04 So we're going to have to scale that down and say there are 12 orders in 60 minutes, which is one hour, so how does that compare to five minutes? so if we were to cross -multiply here or simplify that fraction, we're going to find that our lambda is going to be one order every five minutes.

01:39 So in part a, we want the probability that we have no orders in the five minutes.

01:47 So that would be x would equal zero.

01:51 So we're going to substitute our values in.

01:54 So we will have e to the negative lambda, which we said was one, times lambda to the x power over x factorial.

02:15 Now from your algebra days, anything to the zero power is one, and zero factorial is one.

02:22 So this would be e to the the negative one power or approximately 0 .367879 -4 -4 -1 -2.

02:37 Now, depending on your teacher or your professor, you're going to probably round that.

02:43 I tend to go to four decimal places.

02:47 So i would say 0 .367.

02:51 So we're ready to do part b.

02:56 In part b, we have to think about the concept that poisson probabilities are discrete in nature, and we are looking for the probability of three or more orders.

03:13 So that would be x is greater than or equal to three.

03:17 So in order to do that, we're going to be thinking of our discrete probability distribution, which is a table.

03:24 And our probability could be found that we had no orders or one order or two orders or three orders or four orders.

03:35 And the problem with this is that table never ends.

03:40 So what you've got to think about is the concept of the discrete probability distribution will have a sum of probabilities equivalent to one.

03:53 So we used the formula to find zero, and we found that to be 0 .367, 8, 7, 9, 4, 4, 1, 2.

04:06 If we use that formula and we replace all the xs with a 1, so we would be replacing here with a 1 and here with a 1, we're going to get a value of 0 .36789441 .2.

04:30 And if we replace all those xs with a 2, we will get 0 .1839 -39 -39 -7 -2.

04:45 So i want you to think about the fact that we're going to break this chart into two different parts.

04:51 Could say that the probability that x is less than or equal to two plus the probability that x is greater than or equal to three has to equal one meaning the top half of the chart plus the bottom half of the chart has to equal one so if i were to subtract the probability that x was less than or equal to two from both sides, then i have the statement that the probability that x is greater than or equal to three equals one minus the probability that x is less than or equal to two.

05:34 And if i were to sum up all of these values, i would have the probability that x was less than or equal to 2...

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