5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
2:46 minutesProblem 4.3.12
Textbook Question
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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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 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.
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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.
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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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