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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
5장, 문제 5.R.10b

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).


b. Find the probability that on a given day, there are no deaths.

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1
Step 1: Recognize that this is a Poisson probability problem. The Poisson distribution is used to model the number of events (e.g., deaths) occurring in a fixed interval of time or space, given a known average rate (λ). Here, the average rate is 7 deaths per year.
Step 2: Convert the average rate (λ) from yearly to daily. Since there are 365 days in a year, divide the yearly rate by 365 to find the daily rate: λ = 7 / 365.
Step 3: Use the Poisson probability formula to calculate the probability of 0 deaths on a given day. The formula is: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (in this case, k = 0), λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).
Step 4: Substitute k = 0 and the daily rate λ (calculated in Step 2) into the formula. Simplify the expression: P(X = 0) = (λ^0 * e^(-λ)) / 0!. Note that 0! = 1 and λ^0 = 1, so the formula simplifies to P(X = 0) = e^(-λ).
Step 5: Compute e^(-λ) using the daily rate λ. This will give you the probability of no deaths occurring on a given day.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as deaths in a small population, where the events occur independently of each other.
추천 영상:
가이드 코스
06:38
Intro to Frequency Distributions

Rate Parameter (λ)

In the context of the Poisson distribution, the rate parameter (λ) represents the average number of occurrences in the specified interval. For the village of Westport, with an average of 7 deaths per year, λ would be 7. When calculating probabilities for shorter intervals, such as a day, λ must be adjusted accordingly (e.g., λ = 7/365 for daily calculations).
추천 영상:
가이드 코스
05:53
Parameters vs. Statistics

Probability of No Events

To find the probability of observing no events in a Poisson distribution, the formula P(X=0) = e^(-λ) is used, where e is the base of the natural logarithm. This formula indicates the likelihood of zero occurrences when the average rate is λ. In this case, it helps determine the probability of no deaths occurring on a given day in Westport.
추천 영상:
가이드 코스
05:54
Probability of Multiple Independent Events
관련 실천
교과서 질문

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