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Chapter 5: Problem 7

Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is \(6.1\) peryear, as in Example 1; and proceed to find the indicated probability. Hurricanes a. Find the probability that in a year, there will be 7 hurricanes. b. In a 55 -year period, how many years are expected to have 7 hurricanes? c. How does the result from part (b) compare to the recent period of 55 years in which 7 years had 7 hurricanes? Does the Poisson distribution work well here?

Short Answer

Expert verified
a. \(0.0595\). b. \(3.27\) years. c. The Poisson distribution does not fit well as the actual number of years is more than double the expected.

Step by step solution

01

Understand the Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. The probability mass function of a Poisson random variable is given by: \[ P(X = k) = \frac{{\lambda^k e^{-\lambda}}}{{k!}} \] where \( \lambda \) is the mean number of events, \( k \) is the number of events we want to find the probability for, and \( e \) is the base of the natural logarithm.
02

Identify Given Values for Part (a)

Given that the mean number of hurricanes \( \lambda = 6.1 \) per year and we need to find the probability of 7 hurricanes (\( k = 7 \)).
03

Calculate the Probability Using Poisson Formula

Substitute \( \lambda = 6.1 \) and \( k = 7 \) into the Poisson formula: \[ P(X = 7) = \frac{{6.1^7 e^{-6.1}}}{{7!}} \]. First, calculate each part: \( 6.1^7 = 1329435.7181 \) \( e^{-6.1} \approx 0.0022415 \) \( 7! = 5040 \). Then, \[ P(X = 7) = \frac{{1329435.7181 \times 0.0022415}}{{5040}} \approx 0.0595 \]
04

Compute the Expected Years with 7 Hurricanes for Part (b)

In a 55-year period, the expected number of years with exactly 7 hurricanes can be found by multiplying the probability by the number of years. \[ E = 55 \times P(X = 7) \]. Using the probability from Step 3: \[ E = 55 \times 0.0595 \approx 3.27 \]
05

Compare Results for Part (c)

Compare the expected number of years calculated in Step 4 to the actual number of years with 7 hurricanes (7 years). Expected: \( 3.27 \) years Actual: \( 7 \) years. Since the actual number of years (7) is more than twice the expected number of years (3.27), the Poisson distribution does not fit the observed data well.

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Key Concepts

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

Poisson distribution
The Poisson distribution is crucial in statistics for modeling the number of events within a fixed interval. This distribution assumes that these events happen with a known constant mean rate and independently of the time since the last event.
The formula for the Poisson probability mass function (PMF) is given by: \[ P(X = k) = \frac{{\lambda^k e^{-\lambda}}}{{k!}} \]
Here:
  • \(\lambda\) is the mean number of events (e.g., hurricanes in a year).
  • \(k\) is the number of events we want the probability for.
  • \(e\) is Euler's number (approximately 2.71828).

The Poisson distribution is useful for scenarios where events occur randomly and independently, such as the annual number of hurricanes making landfall.
probability calculation
Calculating the probability in a Poisson distribution involves substituting values into the PMF formula. For example, if the mean number of hurricanes per year is \(\lambda = 6.1\), and we want to find the probability of exactly 7 hurricanes in a year, we substitute these values into the formula: \[ P(X = 7) = \frac{6.1^7 e^{-6.1}}{7!} \]
Breaking it down:
  • First, calculate \(6.1^7\) which gives approximately 1,329,435.718.
  • Next, find \(e^{-6.1}\), approximately 0.002241.
  • Then, determine 7! (7 factorial), which is 5,040.
Combining these parts, we get: \[ P(X = 7) = \frac{1,329,435.718 \times 0.002241}{5,040} \approx 0.0595 \]
Thus, the probability of having exactly 7 hurricanes in a year is approximately 0.0595.
expected value
The expected value in the context of Poisson distribution helps predict how often an event will occur over a larger sample. It's calculated by multiplying the probability of a single event by the number of trials.
For example, if we want to know how many years out of a 55-year period will have exactly 7 hurricanes, we use the probability calculated earlier: \[ E = 55 \times P(X = 7) \]
With \(P(X = 7) \approx 0.0595\), we compute: \[ E = 55 \times 0.0595 \approx 3.27 \]
So, in a span of 55 years, we expect about 3.27 years to have exactly 7 hurricanes.
statistical comparison
Statistical comparison involves comparing expected values with actual observed values to assess a model's accuracy.
In the given problem, we calculated that, based on the Poisson distribution, about 3.27 out of 55 years should have 7 hurricanes. However, the observed data shows that there were 7 years with 7 hurricanes.
Comparing these figures:
  • Expected: 3.27 years
  • Actual: 7 years

Since 7 is significantly higher than 3.27, the Poisson distribution does not fit the observed data well in this case. It suggests more variability than what the Poisson model would predict.
hurricanes data
When analyzing hurricanes' data using the Poisson distribution, it's essential to understand both the historical patterns and the limitations of statistical models.
The mean number of hurricanes, given as 6.1 per year, serves as our base for using the Poisson model. Yet, natural phenomena like hurricanes can have unpredictable variations.
Over the observed 55-year period, more years had 7 hurricanes than the model predicted. This discrepancy highlights the need for cautious interpretation of statistical models. Natural events may not always fit neatly into theoretical distributions due to their inherent unpredictability and the influence of various climatic factors.

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