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Chapter 5: Q11SE (page 344)
Suppose that two random variables \({X_{1\,}}\,and\,\,{X_2}\) have a bivariate normal distribution, and \(Var({X_{1\,}}) = \,Var(\,{X_2})\). Show that the sum\({X_{1\,}}\, + \,\,{X_2}\) and the difference \({X_{1\,}}\, - \,{X_2}\) are independent random variables.
It has been proved.
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Consider again the two tests A and B described in Exercise2. If a student is chosen at random, what is the probability that her score on test A will be higher than her score on test B?
Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}\) are independent and that \({{\bf{X}}_{\bf{i}}}\) has the Poisson distribution with mean \({{\bf{\lambda }}_{\bf{i}}}\left( {{\bf{i = 1, \ldots ,k}}} \right)\). Show that for each fixed positive integer n, the conditional distribution of the random Vector \({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}} \right)\), given that \(\sum\limits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{X}}_{\bf{i}}}{\bf{ = n}}} \) it is the multinomial distribution with parameters n and
\(\begin{array}{l}{\bf{p = }}\left( {{{\bf{p}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{p}}_{\bf{k}}}} \right){\bf{,}}\,{\bf{where}}\\{{\bf{p}}_{\bf{i}}}{\bf{ = }}\frac{{{{\bf{\lambda }}_{\bf{i}}}}}{{\sum\limits_{{\bf{j = 1}}}^{\bf{k}} {{{\bf{\lambda }}_{\bf{j}}}} }}\,{\bf{for}}\,{\bf{i = 1, \ldots ,k}}{\bf{.}}\end{array}\)
LetXhave the lognormal distribution with parameters 3 and 1.44. Find the probability thatX鈮�6.05.
Suppose that two random variables \({{\bf{X}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{X}}_{\bf{2}}}\)have a bivariate normal distribution, and two other random variables \({{\bf{Y}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{Y}}_{\bf{2}}}\)are defined as follows:
\(\begin{array}{*{20}{l}}{{{\bf{Y}}_{\bf{1}}}{\rm{ }} = {\rm{ }}{{\bf{a}}_{{\bf{11}}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{{\bf{12}}}}{{\bf{X}}_{\bf{2}}}{\rm{ }} + {\rm{ }}{{\bf{b}}_{\bf{1}}},}\\{{{\bf{Y}}_{\bf{2}}}{\rm{ }} = {\rm{ }}{{\bf{a}}_{{\bf{21}}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{{\bf{22}}}}{{\bf{X}}_{\bf{2}}}{\rm{ }} + {\rm{ }}{{\bf{b}}_{\bf{2}}},}\end{array}\)
where
\(\left| {\begin{array}{*{20}{l}}{{{\bf{a}}_{{\bf{11}}}}{\rm{ }}{{\bf{a}}_{{\bf{12}}}}}\\{{{\bf{a}}_{{\bf{21}}}}{\rm{ }}{{\bf{a}}_{{\bf{22}}}}}\end{array}} \right|\)
Show that \({{\bf{Y}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{Y}}_{\bf{2}}}\) also have a bivariate normal distribution.
Suppose that\({X_1}and\,{X_2}\) have the bivariate normal distribution with means\({\mu _1}and\,{\mu _2}\) variances\({\sigma _1}^2and\,{\sigma _2}^2\), and correlation 蟻. Determine the distribution of\({X_1} - 3{X_2}\).
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