Chapter 5: Q1SE (page 344)
Consider again the joint distribution of heights of husbands
and wives in Example 5.10.6. Find the 0.95 quantileof the conditional distribution of the height of the wife given that the height of the husband is 72 inches.
The 95th quantile is 70.57.\(\)
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It is said that a random variable X has the Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\,{\bf{and}}\,{\bf{\alpha }}\) if X has a continuous distribution for which the pdf\({\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right)\) is as follows
\(\begin{array}{l}{\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right){\bf{ = }}\frac{{{\bf{\alpha }}{{\bf{x}}_{\bf{0}}}^{\bf{\alpha }}}}{{{{\bf{x}}^{{\bf{\alpha + 1}}}}}}\,{\bf{,x}} \ge {{\bf{x}}_{\bf{0}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{ = }}\,{\bf{0}}\,\,{\bf{,x < }}{{\bf{x}}_{\bf{0}}}\end{array}\)
Show that if X has this Pareto distribution, then the random variable\({\bf{log}}\left( {{\bf{X|}}\,{{\bf{x}}_{\bf{0}}}} \right)\)has the exponential distribution with parameter ฮฑ.
Suppose that a fair coin is tossed until at least one head and at least one tail has been obtained. Let X denote the number of tosses that are required. Find the p.f. of X
Let \({\bf{f}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}} \right)\) denote the p.d.f. of the bivariate normaldistribution specified by Eq. (5.10.2). Show that the maximumvalue of \({\bf{f}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}} \right)\) is attained at the point at which \({{\bf{x}}_{\bf{1}}} = {{\bf{\mu }}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{x}}_{\bf{2}}} = {{\bf{\mu }}_{\bf{2}}}.\)
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 \(\overline X \) ?
Prove Theorem 5.3.3. Hint:Prove that\[\mathop {lim}\limits_{n \to \infty } {c_n}log\left( {1 + {a_n}} \right) - {a_n}{c_n} = 0\]by applying Taylorโs theorem with remainder (see Exercise 13 in Sec. 4.2) to the functionf (x)=log(1+x)aroundx=0.
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