A popular clothing manufacturer receives Internet orders via...

A popular clothing manufacturer receives Internet orders via two different routing systems. The time between orders for each routing system in a typical day is known to be exponentially distributed with a mean of 3.2 minutes. Both systems operate independently. (a) What is the probability that no orders will be received in a 5-minute period? In a 10 -minute period? (b) What is the probability that both systems receive two orders between 10 and 15 minutes after the site is officially open for business? (c) Why is the joint probability distribution not needed to answer the previous questions?

A popular clothing manufacturer receives Internet orders via two different routing systems. The time between orders for each routing system in a typical day is known to be exponentially distributed with a mean of 3.2 minutes. Both systems operate independently.
(a) What is the probability that no orders will be received in a 5-minute period? In a 10 -minute period?
(b) What is the probability that both systems receive two orders between 10 and 15 minutes after the site is officially open for business?
(c) Why is the joint probability distribution not needed to answer the previous questions?

Key Concepts

Exponential Distribution

The exponential distribution is used to model the time between successive independent events occurring at a constant average rate. Its key characteristic is the memoryless property, meaning that the probability of an event occurring in the future is independent of the past. In contexts such as order arrivals where the mean time between orders is known, the exponential distribution allows us to calculate the likelihood of no orders in a given time period.

Poisson Process

A Poisson process is a model for events that occur independently and randomly over time with a constant rate. When the time between events is exponentially distributed, the number of events in a fixed period follows a Poisson distribution. This process is crucial for determining the probability of a certain number of events (such as orders) occurring within a specified interval.

Rate Parameter (?)

The rate parameter, often denoted as ?, represents the average number of events per unit time in a Poisson process. It is the reciprocal of the mean of the exponential distribution. Knowing ? allows one to compute probabilities for both the timing of events and the counting of events in given intervals.

Superposition of Independent Poisson Processes

When there are multiple independent Poisson processes, the total process formed by their combination is also Poisson distributed with a rate equal to the sum of the individual rates. This concept is useful when analyzing simultaneous processes such as two independent routing systems receiving orders, as it simplifies the computation of overall probabilities.

Independence and Joint Distribution Simplification

The independence of processes implies that the events occurring in one process do not affect those in another. This leads to the joint probability distribution being a product of the individual distributions. As a result, detailed joint distributions are unnecessary for computing probabilities for independent events, thereby simplifying the analysis.

Transcript

00:01 Hi guys, this problem from the given information, the time between orders for each routing system and a typical days is exponentially distributed with mean equals 3 .2.

00:12 So we have lambda, which is 1 over the mean.

00:16 Okay, so it's 1 over 3 .2 minutes.

00:20 Okay.

00:21 Now let's compute the probability that no orders will be received in 5 minutes.

00:26 So we need to find the probability of x more than 5 and y more than 5.

00:33 Okay, so it's double integration from 5 to infinity.

00:37 Then from 5 to infinity for 1 over 3 .2 squared times e power negative x over 3 .2 times e power negative x over 3 .2 times e power negative x over 3 .2 times e power negative x over 3 .2 .2 times e power negative x over 3 .2.

00:57 D y d x okay so after integration we get these values as this equals 1 over 32 times e power and negative 1 .5625 times 3 over 3 .2 times e power negative 1 .525 times e 565.

01:33 So this is 0 .0439.

01:38 Okay.

01:41 Then we need to compute the probabilities that no orders will be received in a 10 minutes period.

01:47 Okay.

01:48 So it's the same part.

01:50 Our integration limits starts from 10, not 5.

01:53 So it's probability of x more than 10 and y more than 10.

02:00 Okay.

02:01 So, so.

02:01 So this is integration from 10 to infinity, 10 to infinity for 1 over 3 .2 squared, e power negative x over 3 .2, e power negative y over 3 .2, d, y, d, x.

02:22 Okay, so this after integration, we get this 1 over 3 .2 times e, power negative, 3 .125 times 3 .2 times e power negative 3 .125.

02:43 Okay...

Please give Ace some feedback