91Ó°ÊÓ

Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather conditions. Winter Wind Studies in Rocky Mountain National Park by Glidden (published by Rocky Mountain Nature Association) states that sustained galeforce winds (over 32 miles per hour and often over 90 miles per hour) occur on the average of once every 60 hours at a Trail Ridge Road weather station. (a) Let \(r=\) frequency with which gale-force winds occur in a given time interval. Explain why the Poisson probability distribution would be a good choice for the random variable \(r\) (b) For an interval of 108 hours, what are the probabilities that \(r=2,3,\) and 4? What is the probability that \(r<2 ?\) (c) For an interval of 180 hours, what are the probabilities that \(r=3,4,\) and 5? What is the probability that \(r<3 ?\)

Step by step solution

01

Understanding the Poisson Distribution

The Poisson distribution is suitable for modeling the number of times an event occurs within a specified interval. It is used here to model the frequency of gale-force winds because these wind events are rare and occur independently over time, with a known average rate.

02

Calculating the Average Rate for Different Intervals

The average rate of gale-force winds is once every 60 hours. For a 108-hour interval, the average rate \( \lambda \) is \( \frac{108}{60} = 1.8 \) winds. For a 180-hour interval, \( \lambda \) is \( \frac{180}{60} = 3 \) winds.

03

Calculating Probabilities for Interval of 108 Hours

Using the Poisson probability formula \( P(r=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \):- For \( r=2 \), \( P(2) = \frac{e^{-1.8} \cdot 1.8^2}{2!} \approx 0.270 \).- For \( r=3 \), \( P(3) = \frac{e^{-1.8} \cdot 1.8^3}{3!} \approx 0.162 \).- For \( r=4 \), \( P(4) = \frac{e^{-1.8} \cdot 1.8^4}{4!} \approx 0.073 \).- For \( r<2 \), \( P(r<2) = P(0) + P(1) = \frac{e^{-1.8} \cdot 1.8^0}{0!} + \frac{e^{-1.8} \cdot 1.8^1}{1!} \approx 0.175 + 0.315 = 0.490 \).

04

Calculating Probabilities for Interval of 180 Hours

Using the Poisson probability formula with \( \lambda = 3 \):- For \( r=3 \), \( P(3) = \frac{e^{-3} \cdot 3^3}{3!} \approx 0.224 \).- For \( r=4 \), \( P(4) = \frac{e^{-3} \cdot 3^4}{4!} \approx 0.168 \).- For \( r=5 \), \( P(5) = \frac{e^{-3} \cdot 3^5}{5!} \approx 0.101 \).- For \( r<3 \), \( P(r<3) = P(0) + P(1) + P(2) = \frac{e^{-3} \cdot 3^0}{0!} + \frac{e^{-3} \cdot 3^1}{1!} + \frac{e^{-3} \cdot 3^2}{2!} \approx 0.050 + 0.149 + 0.224 = 0.423 \).

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Key Concepts

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

Probability Calculation

Probability calculation is a core aspect of statistical analysis that helps us understand the likelihood of various outcomes in uncertain situations. In this exercise, we use the Poisson distribution to calculate the probability of gale-force winds occurring over time intervals of 108 and 180 hours. To calculate these probabilities, we use the Poisson probability formula:

• \[ P(r=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]

The formula allows us to find the probability of observing exactly \( k \) number of events given an average rate \( \lambda \). In our scenario, \( \lambda \) equals the expected number of gale-force wind events in a given time interval. Breaking down the calculation:

• \( e \) is the base of the natural logarithm, approximately 2.718.
• \( \lambda \) is the average number of occurrences (events) over the interval.
• \( k! \) is the factorial of \( k \), calculated as the product of all positive integers up to \( k \).

With these elements, we are able to compute the likelihood of specific wind events for the hours studied, making it an essential tool for decision-making in weather forecasting and emergency preparedness.

Random Variables

Random variables are fundamental in the realm of probability and statistics. They are numerical outcomes that result from a random phenomenon. In the context of our problem, the random variable \( r \) represents the frequency of gale-force wind events.Random variables can be of two types:

• Discrete Random Variables: These have countable outcomes, like the number of gale-force wind events which can be 0, 1, 2, etc.
• Continuous Random Variables: These have outcomes that form a continuum, like the exact speed of the wind measured in miles per hour.

In this exercise, \( r \) is a discrete random variable because the number of wind events can only take non-negative whole number values. Understanding random variables helps us quantify and model real-world uncertainties, providing insights into likelihoods and guiding strategic planning in various fields, such as meteorology and risk management.

Statistical Modeling

Statistical modeling involves the use of mathematical frameworks to represent complex phenomena and make informed predictions. The choose of a Poisson distribution as our model in this exercise reflects its efficiency in modeling the frequency of rare, independent events over time. Statistical models are beneficial because:

• They enable prediction by capturing the essence of past data to foresee future occurrences.
• They help in decision-making by estimating the likelihoods of various outcomes.
• They make information digestible by providing a simplified understanding of complex systems.

Through statistical modeling, we can transform data into actionable insights. By incorporating historical measurements—like wind occurrences—we gain a deeper understanding of probable future behavior. Such applications not only aid in effective forecasting but also in crafting plans to mitigate potential risks associated with those forecasts.