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Chapter 5: Problem 50

Let \(x\) be a Poisson random variable. Using the Poisson probabilities table, write the probability distribution of \(x\) for each of the following. Find the mean, variance, and standard deviation for each of these probability distributions. Draw a graph for each of these probability distributions. a. \(\lambda=1.3\) b. \(\lambda=2.1\)

Short Answer

Expert verified
The probability distributions for \(\lambda = 1.3\) and \(\lambda = 2.1\) follow the formula \[P(x=k) = \frac{e^{-\lambda}\lambda^k}{k!}\]. The mean and variance for each are equal to their respective lambda values, with standard deviations approximately 1.14 for \(\lambda = 1.3\) and 1.45 for \(\lambda = 2.1\).

Step by step solution

01

Determining the Poisson Distribution for Lambda 1.3

The probability mass function of a Poisson random variable \(x\) with parameter \(\lambda = 1.3\) is given by: \[P(x=k) = \frac{e^{-\lambda}\lambda^k}{k!}\] where \(e\) is the base of the natural logarithm, \(\lambda\) is the average number of successes that result from the experiment, and \(k\) represents the actual number of successes that result from the experiment.
02

Calculating Mean, Variance, and Standard Deviation for Lambda 1.3

For a Poisson distribution, both the mean and the variance equal \(\lambda\). So, for \(\lambda=1.3\), both mean and variance are 1.3. The standard deviation is the square root of variance, which equals approximately 1.14.
03

Determining the Poisson Distribution for Lambda 2.1

The probability mass function of a Poisson random variable \(x\) with parameter \(\lambda = 2.1\) is given by: \[P(x=k) = \frac{e^{-\lambda}\lambda^k}{k!}\]
04

Calculating Mean, Variance, and Standard Deviation for Lambda 2.1

For this Poisson distribution, both the mean and the variance equal \(\lambda=2.1\). The standard deviation is the square root of the variance, which is approximately 1.45.
05

Drawing the Distributions

Draw two separate graphs for \(\lambda = 1.3\) and \(\lambda = 2.1\). The x-axis is the number of successes \(k\) and the y-axis is the respective probability \(P(x=k)\). The graphs will start at \(k=0\) and continue upwards, the points on the graph will follow the Poisson distribution determined in the previous steps.

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Most popular questions from this chapter

A university police department receives an average of \(3.7\) reports per week of lost student ID cards. a. Find the probability that at most 1 such report will be received during a given week by this police department. Use the Poisson probability distribution formula. b. Using the Poisson probabilities table, find the probability that during a given week the number of such reports received by this police department is i. 1 to 4 ii. at least 6 iii. at most 3

Residents in an inner-city area are concerned about drug dealers entering their neighborhood. Over the past 14 nights, they have taken turns watching the street from a darkened apartment. D? ug deals seem to take place raudomly at vauious times and locations on the street and average about three per night. The residents of this street contacted the local police, who informed them that they do not have sufficient resources to set up surveillance. The police suggested videotaping the activity on the street, and if the residents are able to capture five or more drug deals on tape, the police will take action. Unfortunately, none of the residents on this street owns a video camera and, hence, they would have to. rent the equipment. Inquiries at the local dealers indicated that the best available rate for renting a video camera is \(\$ 75\) for the first night and \(\$ 40\) for each additional night. To obtain this rate, the residents must sign up in advance for a specified number of nights. The residents hold a neighborhood meeting and invite you to help them decide on the length of the rental period. Because it is difficult for them to pay the rental fees, they want to know the probability of taping at least five drug deals on a given number of nights of videotaping. a. Which of the probability distributions you have studied might be helpful here? b. What assumption(s) would you have to make? c. If the residents tape for two nights, what is the probability they will film at least five drug deals? d. For how many nights must the camera be rented so that there is at least . 90 probability that five or more drug deals will be taped?

Let \(N=8, r=3\), and \(n=4\). Using the hypergeometric probability distribution formula, find a. \(P(2)\) b. \(P(0)\) c. \(P(x \leq 1)\)

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(5)\) for \(n=8\) and \(p=.70\) b. \(P(3)\) for \(n=4\) and \(p=.40\) c. \(P(2)\) for \(n=6\) and \(p=.30\) Verify your answers by using Table I of Appendix \(\mathrm{B}\).

The following table lists the probability distribution of the number of patients entering the emergency room during a 1-hour period at Millard Fellmore Memorial Hospital. $$ \begin{array}{l|ccccccc} \hline \text { Patients per hour } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .2725 & .3543 & .2303 & .0998 & .0324 & .0084 & .0023 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution.
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