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Chapter 5: Problem 13

USA Today reported that the U.S. (annual) birth rate is about 16 per 1000 people, and the death rate is about 8 per 1000 people. (a) Explain why the Poisson probability distribution would be a good choice for the random variable \(r=\) number of births (or deaths) for a community of a given population size.

Short Answer

Expert verified
The Poisson distribution is suitable because births and deaths are independent, rare events with a constant average rate within a population.

Step by step solution

01

Identify the scenario

We are discussing the birth and death rates per 1000 people, which are continuous and independent events occurring over a fixed interval (like a year). The events of births and deaths are rare compared to the population size, but they occur independently and their average rate of occurrence is known.
02

Define the random variable

The random variable in question is either the number of births or the number of deaths in a given year for a community, denoted by r. This variable counts the number of these events within a specific population size.
03

Determine characteristics of Poisson distribution

The Poisson distribution is suited to model the number of times an event happens in a fixed interval of time or space. Key characteristics include: events are rare relative to the population, they occur independently, and they occur at a constant average rate.
04

Match scenario to Poisson characteristics

The birth and death events fit the Poisson distribution definition as they are independent, occur at a constant average rate within the population, and are rare relative to the total population size. Thus, the occurrences of these events can be modeled using the Poisson distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Birth Rate
In demographics, birth rate refers to the number of live births occurring during a year per 1,000 people in a given population. It's a critical metric used by demographers to understand and predict population growth.
  • A high birth rate can indicate a growing population, whereas a low birth rate might suggest stagnant or declining population numbers.
  • The calculation is straightforward: divide the number of live births by the population and then multiply the result by 1,000.
The birth rate is often considered alongside other factors such as fertility rate, which measures the number of children a woman is likely to have during her lifetime. By understanding these rates, policymakers can make informed decisions about resource allocation, healthcare, and infrastructure planning. In the context of a Poisson distribution, the birth rate helps establish the average number of occurrences (births) over a fixed period, such as a year.
Death Rate
Death rate, sometimes referred to as mortality rate, measures the number of deaths in a given population during a specific time frame, usually per 1,000 individuals per year.
  • Like birth rate, it is a crucial metric for understanding population dynamics and predicting trends.
  • To calculate it, divide the number of deaths by the population and multiply by 1,000.
A higher death rate could indicate health issues within a population or an aging demographic, while a lower rate might reflect better healthcare and living conditions. In statistical modeling, particularly with the Poisson distribution, the death rate provides the average frequency of the event (deaths) occurring. This average rate is central to applying the Poisson model to predict the likelihood of different numbers of events happening.
Random Variable
In the context of probability and statistics, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. For birth and death rates, the random variable often represents the number of births or deaths observed in a community.
  • The concept of a random variable is essential for assigning numerical values to real-world outcomes.
  • Random variables can be discrete or continuous, though in Poisson distribution, they are generally discrete.
The random variable allows statisticians to model and create probability distributions for various processes, including births and deaths. It helps quantify the likelihood of different outcomes and serves as the foundation for calculating probabilities using distributions like the Poisson model.
Probability Distribution
A probability distribution describes how the values of a random variable are distributed. It provides a mathematical model that quantifies the likelihood of different outcomes.
  • Probability distributions can be either discrete or continuous, with the Poisson distribution being a common model for discrete random variables.
  • It is used to model rare events in large populations, where each event is independent and occurs at a constant rate.
In the case of birth and death rates within a population, the Poisson distribution is highly suitable because it assumes that:
  • Each birth or death is an independent event.
  • The average rate of occurrence (birth or death rate) is consistent over time.
This understanding of probability distributions allows for more precise predictions and analyses of population trends. It helps in assessing the likelihood of various scenarios, such as the number of births or deaths in a particular community over a year.

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Most popular questions from this chapter

In a carnival game, there are six identical boxes, one of which contains a prize. A contestant wins the prize by selecting the box containing it. Before each game, the old prize is removed and another prize is placed at random in one of the six boxes. Is it appropriate to use the binomial probability distribution to find the probability that a contestant who plays the game five times wins exactly twice? Check each of the requirements of a binomial experiment and give the values of \(n, r\), and \(p\).

For a binomial experiment, how many outcomes are possible for each trial? What are the possible outcomes?

Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing \(77 \%\) of the students in Western Civilization each term. Let \(n=1,2,3, \ldots\) represent the number of times a student takes Western Civilization until the first passing grade is received. (Assume the trials are independent.) (a) Write out a formula for the probability distribution of the random variable \(n\). (b) What is the probability that Susan passes on the first try \((n=1) ?\) (c) What is the probability that Susan first passes on the second try \((n=2) ?\) (d) What is the probability that Susan needs three or more tries to pass Western Civilization? (e) What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass? Hint Use \(\mu\) for the geometric distribution and round.

Blood type A occurs in about \(41 \%\) of the population (Reference: Laboratory and Diagnostic Tests, F. Fischbach). A clinic needs 3 pints of type A blood. A donor usually gives a pint of blood. Let \(n\) be a random variable representing the number of donors needed to provide 3 pints of type \(\mathrm{A}\) blood. (a) Explain why a negative binomial distribution is appropriate for the random variable \(n\). Write out the formula for \(P(n)\) in the context of this application. Hint: See Problem \(26 .\) (b) Compute \(P(n=3), P(n=4), P(n=5)\), and \(P(n=6)\). (c) What is the probability that the clinic will need from three to six donors to obtain the needed 3 pints of type A blood? (d) What is the probability that the clinic will need more than six donors to obtain 3 pints of type A blood? (e) What are the expected value \(\mu\) and standard deviation \(\sigma\) of the random variable \(n\) ? Interpret these values in the context of this application.

Suppose we have a binomial experiment with 50 trials, and the probability of success on a single trial is \(0.02\). Is it appropriate to use the Poisson distribution to approximate the probability of two successes? Explain.
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