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Chapter 7: Q7.3-12E (page 406)

Suppose that the time in minutes required to serve a customer at a certain facility has an exponential distribution for which the value of the parameter 胃is unknown and that the prior distribution of胃is a gamma distributionfor which the mean is 0.2 and the standard deviation is 1. If the average time required to serve a random sample of 20 customers is observed to be 3.8 minutes, what is the posterior distribution of胃?

Short Answer

Expert verified

The posterior distribution is gamma with parameters are 20.04 and 76.2

Step by step solution

01

Given information

The time in minutes required to serve a customer at a certain facility has an exponential distribution for which the value of the parameter胃is unknown and the prior distribution of胃is a gamma distribution for which the mean is 0.2 and the standard deviation is 1.

02

Finding the posterior distribution of \({\bf{\theta }}\)

Let\(\alpha \,and\,\beta \)denote the parameters of the prior gamma distribution of\(\theta \).

Then,\(\alpha /\beta = 0.2\,\,{\rm{and}}\,\,{\alpha ^2}/\beta = 1\)

Therefore,\(\alpha = 0.04\,\,{\rm{and}}\,\,\beta = 0.2\)

Furthermore, the total time period required to serve the sample of 20 customers is\(y = 20 \times 3.8 = 76\)

Therefore, by Theorem 7.3.4, the posterior distribution of\(\theta \)is the gamma distribution for which the parameters are\(\alpha + n\,\,{\rm{and}}\,\,\beta + \sum\limits_{i = 1}^n {{x_i}} \)

Suppose thatX1, . . . ,Xnform a random sample from the exponential distributionwith parameter胃 >0 that is unknown. Suppose also that the prior distribution of胃is the gamma distribution with parameters伪 >0 and尾 >0. Then the posteriordistribution of胃given thatXi=xi(i=1, . . . , n) is the gamma distribution with parameters.

\(\alpha + n\,\,{\rm{and}}\,\,\beta + \sum\limits_{i = 1}^n {{x_i}} \)

\({\rm{i}}{\rm{.e}},\alpha + n\, = 0.04 + 20 = 20.04\)

\({\rm{i}}{\rm{.e}},\,\beta + \sum\limits_{i = 1}^n {{x_i}} = 0.2 + 76 = 76.2\)

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Most popular questions from this chapter

Question: Suppose that a scientist desires to estimate the proportionp of monarch butterflies that have a special typeof marking on their wings.

a. Suppose that he captures monarch butterflies one ata time until he has found five that have this specialmarking. If he must capture a total of 43 butterflies,what is the M.L.E. of p?

b. Suppose that at the end of a day the scientist hadcaptured 58 monarch butterflies and had found onlythree with the special marking. What is the M.L.E.of p?

Suppose that a random sample of size n is taken from an exponential distribution for which the value of the parameter 胃 is unknown, the prior distribution of 胃 is a specified gamma distribution, and the value of 胃 must be estimated by using the squared error loss function. Show that the Bayes estimators, for n = 1, 2,..., form a consistent sequence of estimators of 胃.

Question: Suppose that a certain large population contains k different types of individuals (k 鈮� 2), and let \({{\bf{\theta }}_{\bf{i}}}\)denote the proportion of individuals of type i, for i = 1,...,k. Here, 0 鈮� \({\theta _i}\)鈮� 1 and\({{\bf{\theta }}_{\bf{1}}}{\bf{ + }}...{\bf{ + }}{{\bf{\theta }}_{\bf{k}}}{\bf{ = 1}}\). Suppose also that in a random sample of n individuals from this population, exactly ni individuals are of type i, where\({{\bf{n}}_{\bf{1}}}{\bf{ + }}...{\bf{ + }}{{\bf{n}}_{\bf{k}}}{\bf{ = n}}\). Find the M.L.E.鈥檚 of \({{\bf{\theta }}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{\theta }}_{\bf{n}}}\)

Question: Let \({{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}\) be distinct numbers. Let Y be a discrete random variable with the following p.f.:

\(\begin{array}{c}{\bf{f}}\left( {\bf{y}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\,{\bf{if}}\,{\bf{y}} \in \left\{ {{{\bf{x}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{x}}_{\bf{n}}}} \right\}\\ = 0\,otherwise\end{array}\)

Prove that Var(Y ) is given by Eq. (7.5.5).

Question : Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from a distribution for which the p.d.f. f (x|胃 ) is as follows:

\({\bf{f}}\left( {{\bf{x|\theta }}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}{{\bf{e}}^{{\bf{ - }}\left| {{\bf{x - \theta }}} \right|}}\,\,\,\,{\bf{, - }}\infty {\bf{ < x < }}\infty \)

Also, suppose that the value of 胃 is unknown (鈭掆垶 < 胃 < 鈭�). Find the M.L.E. of 胃.
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