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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

Let \(x\) be a Poisson random variable with mean \(\mu=2.5 .\) Use Table 2 in Appendix I to calculate these probabilities: a. \(P(x \geq 5)\) b. \(P(x<6)\) c. \(P(x=2)\) d. \(P(1 \leq x \leq 4)\)

Teen Magazines Although teen magazines Teen People, Hachette Filipacche, and Elle Girl folded in \(2006,70 \%\) of people in a phone-in poll said teens are still a viable market for print, but they do not want titles that talk to them like they are teens. \({ }^{8}\) They read more sophisticated magazines. A sample of \(n=400\) people are randomly selected. a. What is the average number in the sample who said that teenagers are still a viable market for print? b. What is the standard deviation of this number? c. Within what range would you expect to find the number in the sample who said that there is a viable market for teenage print? d. If only 225 in a sample of 400 people said that teenagers are still a viable market for print, would you consider this unusual? Explain. What conclusions might you draw from this sample information?

Find the mean and standard deviation for a binomial distribution with these values: a. \(n=1000, p=.3\) b. \(n=400, p=.01\) c. \(n=500, p=.5\) d. \(n=1600, p=.8\)

Records show that \(30 \%\) of all patients admitted to a medical clinic fail to pay their bills and that eventually the bills are forgiven. Suppose \(n=4\) new patients represent a random selection from the large set of prospective patients served by the clinic. Find these probabilities: a. All the patients' bills will eventually have to be forgiven. b. One will have to be forgiven. c. None will have to be forgiven.

\(P(x \leq k)\) in each case: a. \(n=20, p=.05, k=2\) b. \(n=15, p=.7, k=8\) c. \(n=10, p=.9, k=9\)
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