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Chapter 5: Problem 58

Under what conditions can the Poisson random variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?

Short Answer

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Question: Under what conditions can a Poisson distribution be used to approximate a binomial distribution, and provide an example of an application of the Poisson distribution other than estimating binomial probabilities. Answer: A Poisson distribution can be used to approximate a binomial distribution when the number of trials (n) is large, the probability of success (p) is small, and the product of n and p is finite but not too large. Generally, if n 鈮� 20 and p 鈮� 0.05, the Poisson distribution can be used as an approximation. An example of an application of the Poisson distribution is modeling the number of phone calls received at a call center in a specific time frame.

Step by step solution

01

Conditions for Poisson Approximation to Binomial Distribution

The Poisson distribution can be used to approximate the binomial distribution probabilities under the following conditions: 1. The number of trials (n) in the binomial distribution is large. 2. The probability of success (p) in the binomial distribution is small. 3. The product of n and p, that is np (mean), should be finite and not too large. Usually, a common rule of thumb states that if n 鈮� 20 and p 鈮� 0.05, then the Poisson distribution can be used as an approximation.
02

Poisson Approximation Formula

The Poisson distribution's probability mass function is given by the formula: P(X=k) = ( e^(-位) * 位^k ) / k! , k = 0,1,2,... Where X is the Poisson random variable, k is the number of events, 位 is the mean (which is the product of n and p from the binomial distribution), and e is the base of the natural logarithm (approximately 2.71828).
03

Other Applications of Poisson Distribution

Apart from estimating certain binomial probabilities, the Poisson distribution has various other real-world applications. Some of the common applications include: 1. Modeling the number of phone calls received at a call center in a specific time frame. 2. Predicting the number of goals scored in a sports match. 3. Estimating the number of accidents at an intersection in a given period. 4. Analyzing the number of printing errors on a book's page. These applications generally involve events occurring independently over time or space at an average rate, with no concern regarding the exact time or location of each event.

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Most popular questions from this chapter

In \(2006,\) the average combined SAT score (reading \(+\) verbal \(+\) writing) for collegebound students in the United States was 1518 (out of 2400). Suppose that approximately \(45 \%\) of all high school graduates took this test, and that 100 high school graduates are randomly selected from throughout the United States. \({ }^{1}\) Which of the following random variables has an approximate binomial distribution? If possible, give the values for \(n\) and \(p\). a. The number of students who took the SAT b. The scores of the 100 students on the SAT

A jar contains five balls: three red and two white. Two balls are randomly selected without replacement from the jar, and the number \(x\) of red balls is recorded. Explain why \(x\) is or is not a binomial random variable. (HINT: Compare the characteristics of this experiment with the characteristics of a binomial experiment given in this section.) If the experiment is binomial, give the values of \(n\) and \(p\).

According to the Humane Society of the United States, there are approximately 65 million owned dogs in the United States, and approximately \(40 \%\) of all U.S. households own at least one dog. \({ }^{4}\) Suppose that the \(40 \%\) figure is correct and that 15 households are randomly selected for a pet ownership survey. a. What is the probability that exactly eight of the households have at least one dog? b. What is the probability that at most four of the households have at least one dog? c. What is the probability that more than 10 households have at least one dog?

Suppose that one out of every ten homeowners in the state of California has invested in earthquake insurance. If 15 homeowners are randomly chosen to be interviewed, a. What is the probability that at least one had earthquake insurance? b. What is the probability that four or more have earthquake insurance? c. Within what limits would you expect the number of homeowners insured against earthquakes to fall?

A psychiatrist believes that \(80 \%\) of all people who visit doctors have problems of a psychosomatic nature. She decides to select 25 patients at random to test her theory. a. Assuming that the psychiatrist's theory is true, what is the expected value of \(x\), the number of the 25 patients who have psychosomatic problems? b. What is the variance of \(x\), assuming that the theory is true? c. Find \(P(x \leq 14)\). (Use tables and assume that the theory is true.) d. Based on the probability in part \(\mathrm{c}\), if only 14 of the 25 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist's theory? Explain.
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