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Chapter 5: Problem 84

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. \(\lambda=.6\) b. \(\lambda=1.8\)

Short Answer

Expert verified
For \(\lambda=.6\) and \(\lambda=1.8\), the mean, variance and standard deviation will be .6 and 1.8 respectively. The probability distributions and their graphical representation will depend on the value of \(\lambda\), with the exact calculations as explained in the steps above.

Step by step solution

01

Calculate probabilities

The probability mass function of a Poisson distribution is given by:\[P(X=k) = e^{-\lambda} \frac{\lambda^k}{k!}\]Apply this formula for different values of \(k\) (0,1,2,...) and until the values are negligibly small. Do this for the two different given values of \( \lambda \), i.e, \(.6\) and \(1.8\).
02

Calculate mean, variance, and standard deviation

For a Poisson distribution the mean, variance, and standard deviation are all simple functions of \(\lambda\). The mean and variance of a Poisson distribution are both equal to \( \lambda \). The standard deviation is the square root of the mean and hence also \(\lambda\). So for \(\lambda = .6\) mean, variance and std. deviation will be .6 and for \(\lambda = 1.8\), they will be 1.8.
03

Graph the distribution

Plot each \(P(X=k)\) against \(k\) to display the distribution. This will be a discrete binomial-like distribution but the exact shape will be determined by the value of \(\lambda\). The graph will show on the x-axis the value of the random variable X and on the y-axis the corresponding Poisson probability value \(P(X=k)\). Create one graph for each value of \(\lambda\).

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Most popular questions from this chapter

Which of the following are binomial experiments? Explain why. a. Rolling a die many times and observing the number of spots b. Rolling a die many times and observing whether the number obtained is even or odd c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when \(54 \%\) of all voters are known to be in favor of this proposition.

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