Problem 131 - Applied Statistics and Probability for Engineers

Question

Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 3.5 days. (a) What is the probability that the next change occurs in less than 2.0 days? (b) What is the probability that the time until the next change is greater 7.0 days? (c) What is the time of the next change that is exceeded with probability \(90 \%\) ? (d) What is the probability that the next change occurs in less than 10.0 days, given that it has not yet occurred after 3.0 days?

Step-by-Step Solution

Answer

(a) 0.4146, (b) 0.1353, (c) 36.05 days, (d) 0.7225

Step 1: Understanding the Exponential Distribution

To solve these problems, we first need to understand that the Poisson process implies that the time between events follows an exponential distribution. This distribution is characterized by the rate parameter, \( \lambda \), which is the reciprocal of the mean time between events. For this problem, we have a mean of 3.5 days, so:\[\lambda = \frac{1}{3.5} = \frac{2}{7} \text{ days}^{-1}\]

Step 2: Probability for Less Than 2 Days (Part a)

The probability that the time until the next change occurs in less than 2 days is given by the cumulative distribution function (CDF) of the exponential distribution:\[P(T < 2) = 1 - e^{-\lambda \cdot t} = 1 - e^{-\frac{2}{7} \cdot 2}\]Calculate:\[P(T < 2) = 1 - e^{-\frac{4}{7}} \approx 0.4146\]

Step 3: Probability for Greater Than 7 Days (Part b)

The probability that the next change is greater than 7 days is:\[P(T > 7) = e^{-\lambda \cdot 7} = e^{-\frac{2}{7} \cdot 7} = e^{-2}\]Calculate:\[P(T > 7) \approx 0.1353\]

Step 4: Time Exceeded with 90% Probability (Part c)

We need to find the time, \( t \), such that the probability of exceeding \( t \) is 90%:\[P(T > t) = 0.90 \Rightarrow e^{-\lambda \cdot t} = 0.90\]Solving for \( t \):\[t = -\frac{\ln(0.90)}{\lambda} = -\frac{\ln(0.90)}{\frac{2}{7}}\]Calculate:\[t \approx 36.05 \text{ days}\]

Step 5: Conditional Probability for Less Than 10 Days After 3 Days (Part d)

The conditional probability that the change occurs in less than 10 days given it has not occurred after 3 days is found using:\[P(T < 13 \mid T > 3) = \frac{P(3 < T < 13)}{P(T > 3)}\]We calculate each component:\[P(T > 3) = e^{-\frac{2}{7} \cdot 3} \approx e^{-\frac{6}{7}}\]\[P(T < 13) = 1 - e^{-\frac{2}{7} \cdot 13}\]\[P(3 < T < 13) = P(T < 13) - P(T > 3)\]Finally:\[\frac{(1 - e^{-\frac{26}{7}}) - (e^{-\frac{6}{7}})}{e^{-\frac{6}{7}}}\]Calculate:\[\approx 0.7225\]

Key Concepts

Poisson Process

A Poisson process is often used to model random events that happen over time, such as changes on a website. It's named after the French mathematician Siméon Denis Poisson. This process assumes that these events occur independently and at a constant average rate. As a result, the intervals between events follow an exponential distribution. This means the time between each event (like a change) is random but follows a specific mathematical pattern.

In our exercise, the Poisson process is applied to website updates. With a mean of 3.5 days between changes, web crawlers can estimate how often a site’s content is refreshed. The concept is crucial because knowing these random intervals helps maintain up-to-date information efficiently.

• Models random events over time.
• Events occur independently and at a constant rate.
• Inter-event times follow an exponential distribution.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is a useful tool when working with the exponential distribution in the context of a Poisson process. It represents the probability that a random variable, such as the time until the next website change, will take on a value less than or equal to a specific time.

The CDF for the exponential distribution is expressed as: \[ P(T < t) = 1 - e^{-\lambda \cdot t} \] Where: - \( T \) is the time until the next event - \( \lambda \) is the rate parameter, which is \( \frac{1}{3.5} \) days in our example

This measurement allows us to answer questions such as: "What is the probability that the next change occurs in less than 2 days?" By using the CDF, the probability can be calculated quickly, providing insight into the timing of future events.

• Represents the probability of a variable being less than or equal to a value.
• Helps estimate the probability of time intervals in events.

Conditional Probability

Conditional probability is about finding the likelihood of an event happening given that another event has already occurred. It becomes critical when you're dealing with situations where you have some initial information.

In the context of our exercise, the conditional probability was used to determine the chance of a change happening in less than 10 days, given that it hasn't occurred in the first 3 days. The formula for this kind of probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

• \( A \) is the event that the change happens before 13 days.
• \( B \) is the event that the change hasn't happened after 3 days.

This means we have to calculate \( P(3 < T < 13) \) and \( P(T > 3) \), and then use these to find the desired probability. Understanding this is key as it extends our ability to predict future event timings under specific conditions.

Rate Parameter

The rate parameter, denoted as \( \lambda \), is central to understanding the exponential distribution and, by extension, the Poisson process. It defines how often the random events, like website changes, occur.

In the exponential distribution, \( \lambda \) is the reciprocal of the mean of the distribution. So, if on average, the events occur every 3.5 days, then \( \lambda \) is calculated as: \[ \lambda = \frac{1}{3.5} \text{ days}^{-1} \] This gives us \( \lambda = \frac{2}{7} \) or approximately 0.2857 days\(^{-1}\). The larger the rate parameter, the more frequently events happen. Thus, understanding \( \lambda \) allows us to gain insights into the pacing of events and efficiently predict future occurrences.

• Determines how frequently events occur.
• Is the inverse of the mean in the exponential distribution.
• A higher \( \lambda \) means more frequent events.