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 3

Chapter 5, Problem 3

For the distribution described in Exercise 1, find the probability of exactly 2 arrivals in one thousandth of a minute.

Verified step by step guidance

Identify the type of probability distribution: Since the problem involves counting the number of arrivals in a fixed interval of time, this is a Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when the events occur independently and at a constant average rate.

Write down the Poisson probability mass function (PMF): The PMF is given by P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of arrivals in the given interval, k is the number of arrivals, and e is the base of the natural logarithm (approximately 2.718).

Determine the value of λ: λ represents the expected number of arrivals in the interval. If the average rate of arrivals per minute is given in Exercise 1, multiply that rate by the length of the interval (one thousandth of a minute) to calculate λ.

Substitute k = 2 and the calculated λ into the PMF formula: Replace k with 2 and λ with the value you calculated in the previous step. The formula becomes P(X = 2) = (λ^2 * e^(-λ)) / 2!.

Simplify the expression: Compute the factorial of 2 (2! = 2), raise λ to the power of 2, and multiply by e^(-λ). Divide the result by 2 to find the probability of exactly 2 arrivals in one thousandth of a minute.

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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 arrivals or occurrences, in a defined period. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of occurrence, k is the number of events, and e is Euler's number.

Rate of Arrival (λ)

In the context of the Poisson distribution, λ (lambda) represents the average rate of arrivals or occurrences in a specified time frame. It is a crucial parameter that determines the shape of the distribution. For example, if we expect an average of 5 arrivals in one minute, λ would be 5. When calculating probabilities for shorter intervals, such as one thousandth of a minute, λ must be adjusted accordingly to reflect the expected number of arrivals in that brief period.

Probability Mass Function (PMF)

The Probability Mass Function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. In the case of the Poisson distribution, the PMF allows us to calculate the probability of observing a specific number of events, such as exactly 2 arrivals. By substituting the appropriate values of k and λ into the PMF formula, we can determine the likelihood of that event occurring within the defined time interval.

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Related Practice

Textbook Question

Stem Cell Survey In a Newsweek poll of 882 adults, 481 (or 55%) said that they were in favor of using federal tax money to fund medical research using stem cells obtained from human embryos. A politician claims that people don’t really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Use the following probabilities related to determining whether the result of 481 is significantly high (assuming the true rate is 50%). Is 481 significantly high? What should be concluded about the politician’s claim? Explain.

P(respondent says to use the federal tax money) = 0.5

P(among 882, exactly 481 says to use federal tax money) = 0.000713

P(among 882,481 or more say to use federal tax money) = 0.00389

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Textbook Question

40% of consumers believe that cash will be obsolete in the next 20 years (based on a survey by J.P. Morgan Chase). In each of Exercises 15–20, assume that 8 consumers are randomly selected. Find the indicated probability.

Find the probability that exactly 6 of the selected consumers believe that cash will be obsolete in the next 20 years.

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Textbook Question

Internet Traffic Data Set 27 “Internet Traffic” includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?

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Textbook Question

In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.

Pew Survey In a Pew Research Center survey of 3930 subjects, the ages of the respondents are recorded.

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