/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 The number of views of a page on... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute. a. What is the probability of no views in a minute? b. What is the probability of two or fewer views in 10 minutes? c. Does the answer to the previous part depend on whether the 10 -minute period is an uninterrupted interval? Explain. d. Determine the length of a time interval such that the probability of no views in an interval of this length is \(0.001 .\)

Short Answer

Expert verified
a) The probability is approximately 0.223. b) The probability is small. c) No, it doesn't depend on the interval being continuous. d) The interval is approximately 4.6 minutes.

Step by step solution

01

Understanding the Poisson Distribution

The Poisson distribution is used to model the number of times an event happens within a fixed interval of time or space. Given a mean rate \( \lambda \) (in this case, 1.5 views per minute), we can calculate probabilities using the Poisson probability mass function (PMF): \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where \( X \) is the random variable representing the number of events, \( k \) is a specific value, \( e \) is the base of the natural logarithm (approximately 2.71828), and \( \lambda \) is the mean number of events (1.5 in this case).
02

Probability of No Views in a Minute

For part (a), we need to find the probability of no views (\( k = 0 \)) in one minute where the mean \( \lambda \) is 1.5. Use the Poisson PMF: \[ P(X = 0) = \frac{1.5^0 \times e^{-1.5}}{0!} = e^{-1.5} \] Calculate this value using a calculator.
03

Probability of Two or Fewer Views in 10 Minutes

For part (b), adjust the mean \( \lambda \) for the 10 minute interval: \[ \lambda_{10} = 1.5 \times 10 = 15 \]We need the probability of \( 0, 1, \) or \( 2 \) views. Calculate it by summing the respective Poisson probabilities:\[ P(X \leq 2) = e^{-15} \left( \frac{15^0}{0!} + \frac{15^1}{1!} + \frac{15^2}{2!} \right) \] Calculate each component and sum them.
04

Does the Interval Impact the Probability?

For part (c), the Poisson distribution deals with the number of events in a fixed period, regardless of whether the interval is continuous or split up. Thus, the probability does not depend on whether the time is continuous or interrupted.
05

Length of Interval for Zero Views Probability

For part (d), we need to find the time \( t \) such that the probability of no views is 0.001. Start with:\[ e^{-1.5 \times t} = 0.001 \]Taking natural logarithms on both sides gives:\[ -1.5t = \ln(0.001) \]Solve for \( t \):\[ t = \frac{\ln(0.001)}{-1.5} \] Compute this value using a calculator.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Mass Function
The Probability Mass Function (PMF) is fundamental to understanding the Poisson distribution. It provides the probability of a given number of events happening in a fixed interval of time or space. The Poisson PMF is defined as:
  • \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
  • \( \lambda \) represents the average number of events in the given interval.
  • \( k \) is the number of times the event occurs.
In our example, the PMF helps calculate the probability of receiving a specific number of views on a webpage within a set time frame. Understanding how to apply this function is crucial as it allows predictions about events that occur randomly but follow a certain distribution pattern.
Event Rate Calculation
Event rate calculation is the process of determining the average number of times an event occurs within a given period. For the Poisson distribution, this calculation is essential for using the probability mass function.
  • The event rate is denoted by \( \lambda \), which is the mean occurrence rate (e.g., 1.5 views per minute).
  • Adjustments for different intervals can be made by multiplying \( \lambda \) by the number of intervals, such as \( \lambda_{10} = 15 \) for a 10-minute interval.
This step ensures the PMF accurately reflects different interval lengths. Adapting \( \lambda \) allows calculations for varying conditions while maintaining the integrity of predictions.
Fixed Interval Analysis
Fixed interval analysis is a key feature of the Poisson distribution, allowing the examination of how often an event occurs within a specific timeframe. The concept relates directly to whether probabilities change based on whether intervals are continuous or interrupted.
With Poisson, it's assumed the number of events in non-overlapping intervals is independent. Therefore, the result remains the same whether the interval is broken up or uninterrupted. This is crucial for determining event probabilities over time and ensures consistency in predictions regardless of interval structure.
Natural Logarithms
Natural logarithms play a crucial role in solving equations that arise from Poisson distribution problems, such as determining the interval length for certain probabilities.
  • The natural logarithm, \( \ln \), represents the power to which \( e \) (approximately 2.71828) must be raised to obtain a given number.
  • It facilitates finding unknown variables in exponential equations.
For example, to find how long it takes for the probability of no web page views to be 0.001, you can use the natural logarithm after rearranging the Poisson-derived equation to solve for time. This logarithmic property makes it a powerful tool in exponential scaling scenarios such as these.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

WP The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic panel. Assume that an automobile interior contains 10 square feet of plastic panel. a. What is the probability that there are no surface flaws in an auto's interior? b. If 10 cars are sold to a rental company, what is the probability that none of the 10 cars has any surface flaws? c. If 10 cars are sold to a rental company, what is the probability that at most 1 car has any surface flaws?

Suppose that \(X\) has a hypergeometric distribution with \(N=10, n=3,\) and \(K=4\). Sketch the probability mass function of \(X\). Determine the cumulative distribution function for \(X\).

Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards, and 20 are selected without replacement for functional testing. a. If 20 cards are defective, what is the probability that at least 1 defective card is in the sample? b. If 5 cards are defective, what is the probability that at least 1 defective card appears in the sample?

An article in Knee Surgery, Sports Traumatology, Arthroscopy ["Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique" (2005, Vol. \(13,\) pp. \(273-279\) ) \(]\) cited a success rate of more than \(90 \%\) for meniscal tears with a rim width under \(3 \mathrm{~mm}\), but only a \(67 \%\) success rate for tears of \(3-6 \mathrm{~mm}\). If you are unlucky enough to suffer a meniscal tear of under \(3 \mathrm{~mm}\) on your left knee and one of width \(3-6 \mathrm{~mm}\) on your right knee, what is the probability mass function of the number of successful surgeries? Assume that the surgeries are independent.

3.5 .17 Consider the time to recharge the flash in cell-phone cam eras as in Example \(3.1 .\) Assume that the probability that a camer passes the test is 0.8 and the cameras perform independently What is the smallest sample size needed so that the probability of at least one camera failing is at least \(95 \% ?\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.