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Chapter 5: Problem 73

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
For \(\lambda = 1.3\), the probability distribution, mean, variance, and standard deviation are \(P(X=k) = \frac{e^{-1.3}1.3^k}{k!}\), \(E[X] = 1.3\), \(Var[X] = 1.3\), \(SD[X] = \sqrt{1.3}\) respectively. For \(\lambda = 2.1\), they are \(P(X=k) = \frac{e^{-2.1}2.1^k}{k!}\), \(E[X] = 2.1\), \(Var[X] = 2.1\), \(SD[X] = \sqrt{2.1}\) respectively.

Step by step solution

01

Define Poisson Distribution

Recall that the probability mass function of a Poisson random variable X is given by \(P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}\) where \(k\) is the number of occurrences of an event, and \(\lambda = E[X]\) (the expected value or 'average rate of value') is the parameter defining the distribution.
02

Definitions for lambda = 1.3

When \(\lambda = 1.3\), the probability distribution, mean, variance, and standard deviation are:Probability distribution: \(P(X=k) = \frac{e^{-1.3}1.3^k}{k!}\)Mean: \(\lambda = E[X] = 1.3\)Variance: \(\lambda = Var[X] = 1.3\)Standard Deviation: \(SD[X] = \sqrt{Var[X]} = \sqrt{1.3}\)Hence the probability distribution is dependent on \(k\), the number of occurrences of an event.
03

Definitions for lambda = 2.1

When \(\lambda = 2.1\), the probability distribution, mean, variance, and standard deviation are:Probability Distribution: \(P(X=k) = \frac{e^{-2.1}2.1^k}{k!}\)Mean: \(\lambda = E[X] = 2.1\)Variance: \(\lambda = Var[X] = 2.1\)Standard Deviation: \(SD[X] = \sqrt{Var[X]} = \sqrt{2.1}\)Hence the probability distribution is dependent on \(k\), the number of occurrences of an event.
04

Draw Graphs

To draw the probability distribution graphs, one must plot the probabilities \(P(X=k)\) against \(k\) for each \(\lambda\). The details of graph plotting are better understood through direct visual interpretation rather than through text instructions.

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