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Suppose the number \(X\) of tornadoes observed in a particular region during a 1-year period has a Poisson distribution with \(\lambda=8\). a. Compute \(P(X \leq 5)\). b. Compute \(P(6 \leq X \leq 9)\). c. Compute \(P(10 \leq X)\). d. What is the probability that the observed number of tornadoes exceeds the expected number by more than 1 standard deviation? 81\. Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter \(\lambda=20\) (suggested in the article "Dynamic Ride Sharing: Theory and Practice," J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will a. Be at most 10 ? b. Exceed 20? c. Be between 10 and 20 , inclusive? Be strictly between 10 and 20 ? d. Be within 2 standard deviations of the mean value?

Step by step solution

01

Understanding Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, assuming the events occur with a known constant mean rate and independently of the time since the last event. Here, \( \lambda \) represents the average rate of occurrence.

02

Calculate Probability for Question 1a: \(P(X \leq 5)\)

For a Poisson distribution with \( \lambda = 8 \), the probability that \(X\) is less than or equal to 5 is found using the cumulative distribution function (CDF). \[ P(X \leq 5) = e^{-8} \sum_{x=0}^{5} \frac{8^x}{x!} \]Calculating each term and summing them gives the required probability, approximately 0.191.

03

Calculate Probability for Question 1b: \(P(6 \leq X \leq 9)\)

The probability that \(X\) is between 6 and 9 is calculated by subtracting \( P(X \leq 5) \) from \( P(X \leq 9) \). \[ P(6 \leq X \leq 9) = P(X \leq 9) - P(X \leq 5) \]Calculate \( P(X \leq 9) = e^{-8} \sum_{x=0}^{9} \frac{8^x}{x!} \) and subtract \( P(X \leq 5) \) found previously. After computing, the result is approximately 0.587.

04

Calculate Probability for Question 1c: \(P(10 \leq X)\)

To find \( P(10 \leq X) \), we subtract \( P(X \leq 9) \) from 1. \[ P(10 \leq X) = 1 - P(X \leq 9) \]Using \( P(X \leq 9) \) computed in the previous step, the probability is approximately 0.223.

05

Determine Probability for Question 1d using Standard Deviation

The expected number of tornadoes is \( \lambda = 8 \), and the standard deviation is \( \sqrt{8} \approx 2.83 \).Find \( P(X > 10.83) \), which can be approximated to \( P(X \geq 11) \), using complementary cumulative distribution:\[ P(X \geq 11) = 1 - P(X \leq 10) \]Calculate \( P(X \leq 10) = e^{-8} \sum_{x=0}^{10} \frac{8^x}{x!} \) and subtract from 1, approximately resulting in 0.158.

06

Calculate Probability for Question 2a: \(P(X \leq 10)\)

For a Poisson distribution with \( \lambda = 20 \), we calculate \( P(X \leq 10) \) using the CDF:\[ P(X \leq 10) = e^{-20} \sum_{x=0}^{10} \frac{20^x}{x!} \]Compute the sum. The probability is approximately 0.010.

07

Calculate Probability for Question 2b: \(P(X > 20)\)

To find \( P(X > 20) \), we subtract \( P(X \leq 20) \) from 1:\[ P(X > 20) = 1 - P(X \leq 20) \]Calculate \( P(X \leq 20) = e^{-20} \sum_{x=0}^{20} \frac{20^x}{x!} \), and subtract from 1. The probability is approximately 0.493.

08

Calculate Probability for Question 2c: \(P(10 \leq X \leq 20)\) and strict bounds.

For inclusive bounds:\[ P(10 \leq X \leq 20) = P(X \leq 20) - P(X < 10) \]And for strict bounds, using:\[ P(11 \leq X \leq 19) = P(X \leq 19) - P(X < 11) \]These require computing using the CDF of Poisson by summing terms appropriately.

09

Calculate Probability for Question 2d within 2 Standard Deviations

With \( \lambda = 20 \), the standard deviation is \( \sqrt{20} \approx 4.47 \).Compute probability within two standard deviations \( (11.06 \leq X \leq 28.94) \), which approximates to \(P(11 \leq X \leq 28)\).Using the CDF, compute:\[ P(11 \leq X \leq 28) = P(X \leq 28) - P(X < 11) \]

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Key Concepts

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

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a fundamental concept in probability theory and statistics, especially useful when dealing with Poisson distributions. It gives us the probability that a random variable is less than or equal to a certain value. In essence, the CDF accumulates probabilities from the minimum value up to a particular value of interest. This makes it particularly handy when you're interested in calculating the probability of a range of events.

• The CDF for a Poisson distribution is the sum of probabilities from zero up to the desired number.
• Mathematically, for a Poisson random variable with parameter \( \lambda \), the CDF up to value \( x \) is given by \( \sum_{k=0}^{x} \frac{e^{-\lambda} \lambda^k}{k!} \).

Computing the CDF is the first step in finding the probability of an event in a Poisson distribution because it provides the groundwork for calculating probabilities of ranges of values and more complex inquiries.

Probability Computation

Probability computation in statistics, especially with the Poisson distribution, can initially seem daunting, but it's all about methodical calculation. You assess the likelihood of certain numbers of occurrences within a specified timeframe.With a Poisson distribution:

• To find \( P(X \leq a) \), use the CDF to sum probabilities from \( 0 \) to \( a \).
• For \( P(a \leq X \leq b) \), subtract \( P(X \leq a-1) \) from \( P(X \leq b) \).

To compute \( P(X > c) \), it's helpful to use the complement rule: \( P(X > c) = 1 - P(X \leq c) \). These computations allow students to encompass all challenges seen in Poisson-related problems through different types of probability events.

Standard Deviation in Statistics

Standard deviation is a crucial statistical measure used to quantify the amount of variation or dispersion in a set of data. In the context of the Poisson distribution, this is especially important because it gives you insight into how spread out the values are around the mean.

• For a Poisson distribution, the mean and variance are equal to \( \lambda \).
• The standard deviation is then \( \sqrt{\lambda} \).

This helps to determine the extent to which the actual number of events observed diverges from the expected number. For instance, by knowing the standard deviation, you can assess the probability that the observed number will differ from the mean by more than a certain amount. This is particularly useful in applications where understanding the variability of an event occurrence is critical, like in engineering or natural disaster prediction.

Statistical Modelling in Engineering

Statistical modelling is a vital part of engineering, where decision making often relies on analyzing and interpreting data. The Poisson distribution is frequently used in engineering fields to model random events such as the arrival of customers at a service point, network traffic data, or natural disasters like tornadoes.

• Understanding the expected value and variance allows engineers to predict event outcomes and prepare for various scenarios.
• By leveraging statistical models, you can optimize processes and improve system performance.

Statistical modelling using the Poisson distribution aids in robust design and operational strategies, ensuring safety and efficiency. For example, in transportation engineering, models can predict traffic flow or demand, allowing for better resource allocation and service planning, thus minimizing environmental impact and maximizing system performance.