Chapter 5: Q1E (page 275)
1. Consider a daily lottery as described in Example 5.5.4.
a. Compute the probability that two particular days ina row will both have triples.
b. Suppose that we observe triples on a particular day. Compute the conditional probability that we observe triples again the next day.
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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 \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).
Suppose that seven balls are selected at random withoutreplacement from a box containing five red balls and ten blue balls. If \(\overline X \) denotes the proportion of red balls in the sample, what are the mean and the variance of \(\overline X \) ?
Suppose that X has the normal distribution with mean\({\bf{\mu }}\)and variance\({{\bf{\sigma }}^{\bf{2}}}\). Express\({\bf{E}}\left( {{{\bf{X}}^{\bf{3}}}} \right)\)in terms of\({\bf{\mu }}\)and\({{\bf{\sigma }}^{\bf{2}}}\).
Suppose that the random variables X1 and X2 havea bivariate normal distribution, for which the joint p.d.f.is specified by Eq. (5.10.2). Determine the value of the constant b for which \({\bf{Var}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + b}}{{\bf{X}}_{\bf{2}}}} \right)\)will be a minimum.
Prove Theorem 5.5.5.
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