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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Triola 14th EditionCh. 5 - Discrete Probability Distributions문제 5.3.9b
5장, 문제 5.3.9b

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.


Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.


b. Find the probability that in a single day, there are 16 births.

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Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events (e.g., births) occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The probability mass function is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).
Step 2: Calculate the average number of births per day (λ). Since there are 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.
Step 3: Identify the value of k. In this problem, k represents the number of births in a single day, which is given as 16.
Step 4: Substitute the values of λ and k into the Poisson probability formula. Use the formula P(X = k) = (λ^k * e^(-λ)) / k!. Replace λ with the calculated daily average and k with 16.
Step 5: Simplify the expression to compute the probability. First, calculate λ^k, e^(-λ), and k! (16 factorial). Then, divide the product of λ^k and e^(-λ) by k! to find the probability. This will give you the final result.

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주요 개념

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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 that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.
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Intro to Frequency Distributions

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which gives the expected number of births per day.
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가이드 코스
03:28
Mean & Standard Deviation of Binomial Distribution

Calculating Probability with Poisson

To find the probability of observing exactly k events (in this case, 16 births) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is the base of the natural logarithm, λ is the mean number of events, and k! is the factorial of k. This formula allows us to compute the likelihood of a specific number of occurrences given the average rate.
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07:09
Probability From Given Z-Scores - TI-84 (CE) Calculator
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Using Probabilities for Significant Events


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 In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.


Hurricanes


b. In a 118-year period, how many years are expected to have 7 hurricanes?

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b. In a 118-year period, how many years are expected to have 10 hurricanes?

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