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Chapter 3: Problem 18

The distribution of the time until a Web site changes is important to Web crawlers that search engines use to maintain current information about Web sites. The distribution of the time until change (in days) of a Web site is approximated in the following table. Days until Changes \(\quad\) Probability 1.5 0.05 $$ \begin{array}{ll} 3.0 & 0.25 \\ 4.5 & 0.35 \\ 5.0 & 0.20 \end{array} $$ $$ 7.0 $$ 0.15.

Short Answer

Expert verified
The expected time until a website changes is 4.45 days.

Step by step solution

01

Understand the Problem

We are given a distribution table showing different times (in days) until a website changes content, along with their respective probabilities. This represents a probability distribution for a discrete random variable.
02

Verify the Probability Distribution

Ensure that the sum of all probabilities equals 1. The given probabilities are 0.05, 0.25, 0.35, 0.20, and 0.15. Add these probabilities together to confirm: \( 0.05 + 0.25 + 0.35 + 0.20 + 0.15 = 1.00 \). This confirms it's a valid probability distribution.
03

Calculate the Expected Value

The expected value or mean of a discrete random variable is calculated using the formula: \( E(X) = \sum [x_i \cdot P(x_i)] \), where \( x_i \) represents the time until changes, and \( P(x_i) \) represents the probabilities. Calculate as follows:- \( 1.5 \cdot 0.05 = 0.075 \)- \( 3.0 \cdot 0.25 = 0.75 \)- \( 4.5 \cdot 0.35 = 1.575 \)- \( 5.0 \cdot 0.20 = 1.0 \)- \( 7.0 \cdot 0.15 = 1.05 \) Add these products: \( 0.075 + 0.75 + 1.575 + 1.0 + 1.05 = 4.45 \). The expected value is 4.45 days.

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Key Concepts

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

Random Variables
A random variable is a key concept in probability and statistics, representing a numerical outcome of a random process. In our context, it is used to describe the possibly changing attribute of an event such as the number of days until a website changes its content.

In this exercise, the discrete random variable is 'Days until Website Changes'. It can take on one of a fixed set of values: 1.5, 3.0, 4.5, 5.0, and 7.0.

We call it 'discrete' because the outcomes are specific numerical values that fall within certain intervals, as opposed to 'continuous' random variables that can take any value within a range. Random variables help in modeling real-life scenarios and making informed predictions.
Expected Value
The expected value, often referred to as the mean, is an essential statistical measure that provides the average outcome of a probability distribution if the experiment were repeated many times. It helps us understand where the central tendency of the data lies.

In a discrete probability distribution, its formula is given by:\[ E(X) = \sum [x_i \cdot P(x_i)] \]
where:\
  • \[x_i\] represents each possible value the random variable can take (e.g., days until a website changes)
  • \[P(x_i)\] is the probability of each respective value

For the current exercise, we computed that the expected value of the random variable 'Days until Website Changes' is 4.45 days. Thus, over time and across many websites, you'd expect a change roughly every 4.45 days on average.
Discrete Probability
Discrete probability distributions describe scenarios where the random variables can take on a countable number of different outcomes. The term 'discrete' underscores the fact that we are dealing with distinct, separate values.

For a discrete probability distribution, the sum of all probabilities must equal 1, as each possible outcome has a certain likelihood that cumulatively must cover every possibility.

In our exercise, the probabilities for 'Days until Website Changes' were given as:\[\begin{array}{l}1.5: 0.05 \3.0: 0.25 \4.5: 0.35 \5.0: 0.20 \7.0: 0.15\end{array}\]
These probabilities add up to 1, confirming a valid discrete probability distribution. Understanding this concept helps ensure that probabilistic models align with real-world events and logical reasoning.

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Most popular questions from this chapter

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