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Chapter 6: Problem 38

Do interruptions while you are working reduce your productivity? According to a University of California-Irvine study, businesspeople are interrupted at the rate of approximately \(5^{1 / 2}\) times per hour (Fortune, March 20,2006 ). Suppose the number of interruptions follows a Poisson probability distribution. a. Show the probability distribution for the time between interruptions. b. What is the probability that a businessperson will have no interruptions during a 15 minute period? c. What is the probability that the next interruption will occur within 10 minutes for a particular businessperson?

Short Answer

Expert verified
a. Exponential distribution, b. Probability \( \approx 0.5712 \), c. Probability \( \approx 0.3133 \).

Step by step solution

01

Understanding Poisson Per Hour

We are given that interruptions occur at a rate of \(5^{1/2} = \sqrt{5}\) per hour. This is approximately \(2.236\) interruptions per hour. The Poisson distribution specifies the number of events in a fixed interval. Here, the mean rate (\(\lambda\)) is \(\sqrt{5}\) per hour.
02

Convert Rate to Minutes

Since we need to find probabilities for periods in minutes, we convert the per-hour rate to per-minute. The rate becomes \(\lambda = \frac{\sqrt{5}}{60} \approx 0.03727\) interruptions per minute.
03

Probability Distribution for Exponential

The time between interruptions follows an Exponential distribution with rate parameter \(\lambda\). The probability density function (PDF) is given by \(f(t) = \lambda e^{-\lambda t}\) for \(t \geq 0\). Here, \(\lambda = \sqrt{5}\) (per hour).
04

Probability of No Interruptions in 15 Minutes

For 15 minutes without interruptions, calculate using Poisson. The parameter \(\lambda_{15} = \frac{\sqrt{5}}{4} \approx 0.559\). For no interruptions (\(k=0\)), the probability is \(P(X=0) = e^{-\lambda_{15}} \approx e^{-0.559} \approx 0.5712\).
05

Probability of Next Interruption in 10 Minutes

For an interruption within 10 minutes, use the Exponential CDF. \(P(T < 10) = 1 - e^{-\lambda \cdot 10} = 1 - e^{-\frac{\sqrt{5}}{6}} \approx 1 - e^{-0.373} \approx 0.3133\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Distribution
The Exponential distribution is a continuous probability distribution that describes the time between events in a process where events occur continuously and independently at a constant average rate. In this problem, we're interested in the time between interruptions. Since interruptions follow a Poisson process, the time between them is exponentially distributed.

Key characteristics of the Exponential distribution include:
  • The exponential distribution is defined by its rate parameter, \( \lambda \), which is the average number of events in an interval. In this case, it's related to the number of interruptions.
  • The probability density function (PDF) for the Exponential distribution is \( f(t) = \lambda e^{-\lambda t} \), for \( t \geq 0 \). This formula gives the likelihood of the time until the next interruption.
  • The mean or expected value of an Exponential distribution is \( \frac{1}{\lambda} \).
Understanding the Exponential distribution helps us find how likely it is for certain time intervals to pass before the next interruption occurs, which is central to analyzing work productivity and interruptions.
Probability
Probability is a measure of the likelihood of an event occurring, a core concept in understanding Poisson and Exponential distributions. In this context, probability helps us determine how likely it is that a businessperson experiences interruptions at specific times.

The exponential distribution, for instance, allows us to calculate:
  • The probability of experiencing no interruptions within a certain time frame using the formula for the cumulative distribution function (CDF): \( P(T < t) = 1 - e^{-\lambda t} \)
  • The probability of having exactly \( k \) interruptions within a given period by using the Poisson formula, which serves to help measure interruptions over a specified duration.
For understanding the time intervals and patterns of interruptions, it is crucial to grasp probability concepts, such as calculating events within intervals or none at all.
Rate Conversion
Rate conversion is the process of adapting given rates to suit the time in which you are evaluating an event. In this exercise, the initial interruption rate given is in terms of per hour, but our calculations often contemplate minutes.

Here's how to convert:
  • The original rate is \( \sqrt{5} \) interruptions per hour. To convert to a per-minute rate, divide by 60 (since there are 60 minutes in an hour), giving \( \frac{\sqrt{5}}{60} \approx 0.03727 \) interruptions per minute.
  • This conversion is imperative since we often need probabilities over particular minute spans, like 15 minutes or 10 minutes, rather than for the whole hour.
This adjusted per-minute rate allows us to compute Exponential or Poisson probabilities for specific intervals in a consistent unit, helping analyze the frequency and impact of interruptions on productivity.
Interruption Analysis
Analyzing interruptions involves looking into the frequency and expected duration between events that disrupt work. Using statistical distributions, we measure how disruptions might affect productivity. In this scenario, interruption analysis involves:

  • Using the Poisson distribution to determine how often a certain number of interruptions might occur over a given period, like 15 minutes without any interruptions.
  • Applying the Exponential distribution to find the probability of the next interruption occurring within a specified time, such as 10 minutes.
By understanding these statistical measures, businesses and individuals can assess the impact of interruptions and potentially plan strategies to mitigate their effects on productivity. Analyzing these aspects helps strike a balance between tasks and unexpected disruptions.

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Most popular questions from this chapter

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