Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.3.10

Chapter 5, Problem 5.3.10

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

Murders In a recent year (365 days), there were 650 murders in Chicago. Find the mean number of murders per day, then use that result to find the probability that in a single day, there are no murders. Would 0 murders in a single day be a significantly low number of murders?

Verified step by step guidance

Calculate the mean number of murders per day by dividing the total number of murders (650) by the total number of days in a year (365). Use the formula:

λ = \frac{650}{365}

, where

λ

is the mean.

Recognize that the Poisson distribution is used to model the probability of a given number of events (murders) occurring in a fixed interval of time (1 day) when the events occur independently and at a constant average rate.

To find the probability of 0 murders in a single day, use the Poisson probability formula:

P (X = 0) = \frac{e^{- λ}}{λ^{0}}

, where

e

is the base of the natural logarithm (approximately 2.718),

λ

is the mean, and

X

is the number of events (murders).

Substitute the calculated mean

λ

into the formula to compute

P (X = 0)

. Simplify the expression by noting that

λ^{0} = 1

and

e -^{λ}

is the exponential term.

To determine if 0 murders in a single day is significantly low, compare the probability

P (X = 0)

to a significance threshold (e.g., 0.05). If the probability is less than the threshold, it is considered significantly low.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the number of murders in a day. The distribution is defined by the parameter λ (lambda), which represents the average number of occurrences in the interval.

Mean of the Distribution

In the context of the Poisson distribution, the mean (λ) is the expected number of occurrences in the specified interval. For the given problem, the mean number of murders per day can be calculated by dividing the total number of murders (650) by the number of days (365). This mean value is crucial for determining the probabilities of different outcomes using the Poisson formula.

Significance of Events

Determining whether an event, such as having zero murders in a day, is significantly low involves comparing the observed outcome to the expected distribution. In statistical terms, this often involves calculating the probability of observing such an event under the Poisson distribution and assessing whether this probability falls below a certain threshold (e.g., 0.05), indicating that the event is statistically significant.

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Related Practice

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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