Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

문제 4.3.12

4장, 문제 4.3.12

Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Immigration The mean number of people who immigrated to the United States per hour was about 5.5 in April 2021. Find the probability that the number of people who immigrate to the U.S. in a given hour in April 2021 was (a) zero, (b) exactly five, and (c) exactly eight. (Source: U.S. Census Bureau)

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Step 1: Identify the appropriate probability distribution to use. Since the problem involves the mean number of events (immigrations) occurring in a fixed interval (per hour), and the events are independent, the Poisson distribution is appropriate. The Poisson distribution is defined as P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Define the given parameters. The mean number of immigrations per hour (λ) is 5.5. For each part of the problem, k will represent the specific number of immigrations: (a) k = 0, (b) k = 5, and (c) k = 8.

Step 3: Calculate the probability for part (a). Substitute λ = 5.5 and k = 0 into the Poisson formula: P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Simplify the expression by noting that 0! = 1 and 5.5^0 = 1, so P(X = 0) = e^(-5.5).

Step 4: Calculate the probability for part (b). Substitute λ = 5.5 and k = 5 into the Poisson formula: P(X = 5) = (5.5^5 * e^(-5.5)) / 5!. Simplify the expression by calculating 5.5^5 and 5! (5! = 5 × 4 × 3 × 2 × 1).

Step 5: Calculate the probability for part (c). Substitute λ = 5.5 and k = 8 into the Poisson formula: P(X = 8) = (5.5^8 * e^(-5.5)) / 8!. Simplify the expression by calculating 5.5^8 and 8! (8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1). After finding the probabilities for parts (a), (b), and (c), compare them to a threshold (e.g., 0.05) to determine if the events are unusual.

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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 a known average rate of occurrence. It is particularly useful for modeling rare events, such as the number of immigrants arriving in an hour. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of occurrences, and e is Euler's number.

Mean and Expected Value

The mean, or expected value, of a probability distribution is a measure of the central tendency, representing the average outcome if an experiment were repeated many times. In the context of the Poisson distribution, the mean (λ) indicates the average number of events (e.g., immigrants) expected in a given time frame. Understanding the mean helps in calculating probabilities for specific outcomes, such as zero, five, or eight immigrants in an hour.

Unusual Events

In statistics, an event is often considered unusual if its probability is significantly low, typically below 5%. This threshold helps in identifying outcomes that deviate from what is expected under a given distribution. When analyzing the probabilities of immigration numbers, determining whether the events of zero, five, or eight immigrants are unusual involves comparing their calculated probabilities to this benchmark, providing insight into the likelihood of these occurrences.

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교과서 질문

"Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

The experiment has a fixed number of trials n, where each trial is independent of the other trials.

Each trial has k possible mutually exclusive outcomes:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in Exercise 39 as , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless."

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교과서 질문

In Exercises 5–8, find the indicated probability using the Poisson distribution.

P(3) when μ = 6

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교과서 질문

Discrete Variables and Continuous Variables In Exercises 13–18, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the fitted hat sizes of members of a softball team.

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교과서 질문

Independent and Dependent Random Variables Two random variables x and y are independent when the value of x does not affect the value of y. When the variables are not independent, they are dependent. A new random variable can be formed by finding the sum or difference of random variables. If a random variable x has mean and a random variable y has mean , then the means of the sum and difference of the variables are given by . If random variables are independent, then the variance and standard deviation of the sum or difference of the random variables can be found. So, if a random variable x has variance and a random variable y has variance , then the variances of the sum and difference of the variables are given by In Exercises 43 and 44, the distribution of SAT mathematics scores for college-bound male seniors in 2020 has a mean of 531 and a standard deviation of 121. The distribution of SAT mathematics scores for college-bound female seniors in 2020 has a mean of 516 and a standard deviation of 112. One male and one female are randomly selected. Assume their scores are independent. (Adapted from College Board)

Find the mean and standard deviation of the sum of their scores.

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교과서 질문

Linear Transformation of a Random Variable In Exercises 41 and 42, use this information about linear transformations. For a random variable x, a new random variable y can be created by applying a linear transformation , where a and b are constants. If the random variable x has mean and standard deviation , then the mean, variance, and standard deviation of y are given by the formulas

The mean annual salary of employees at an office is originally \$46,000. Each employee receives an annual bonus of \$600 and a 3% raise (based on salary). What is the new mean annual salary (including the bonus and raise)?

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Constructing and Graphing Binomial Distributions In Exercises 27–30, (a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable x that you would consider unusual. Explain your reasoning.

Workplace Cleanliness Fifty-seven percent of employees judge their peers by the cleanliness of their workspaces. You randomly select 10 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number who judge their peers by the cleanliness of their workspaces. (Source: Adecco)

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