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Chapter 5: Q3E (page 302)
Consider the sequence of coin tosses described in Exercise 2.
a. What is the expected number of tosses that will be required in order to obtain five heads?
b. What is the variance of the number of tosses that will be required in order to obtain five heads?
(a) 150
(b) 4350
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Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 1]. Let \({{\bf{Y}}_{\bf{1}}}{\bf{ = min}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right\}\), \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right\}\)and \({\bf{W = }}{{\bf{Y}}_{\bf{n}}}{\bf{ - }}{{\bf{Y}}_{\bf{1}}}\). Show that each of the random variables \({{\bf{Y}}_{\bf{1}}}{\bf{,}}{{\bf{Y}}_{\bf{n}}}\,\,{\bf{and}}\,\,{\bf{W}}\) has a beta distribution.
The lifetime X of an electronic component has an exponential distribution such that \({\bf{P}}\left( {{\bf{X}}\, \le {\bf{1000}}} \right){\bf{ = 0}}{\bf{.75}}\). What is the expected lifetime of the component?
Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having thenormal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, weshall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).
a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distributionwith both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).
Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).
Nowshow that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normals and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .
b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).
A manufacturer believes that an unknown proportionP of parts produced will be defective. She models P ashaving a beta distribution.The manufacturer thinks that Pshould be around 0.05, but if the first 10 observed productswere all defective, the mean of P would rise from 0.05 to0.9. Find the beta distribution that has these properties.
In Example 5.3.2, compute the probability that all 10 successful patients appear in the subsample of size 11 from the Placebo group.
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