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Chapter 3: Q 11E (page 167)
Suppose that\({X_1}...{X_n}\)are independent. Let\(k < n\)and let\({i_1}.....{i_k}\)be distinct integers between 1 and n. Prove that \(X{i_1}.....X{i_k}\)they are independent.
\(X{i_1}.....X{i_k}\) are independent
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Suppose that three random variables X1, X2, and X3 have a continuous joint distribution with the following joint p.d.f.:
\({\bf{f}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{c}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{ + 2}}{{\bf{x}}_{\bf{2}}}{\bf{ + 3}}{{\bf{x}}_{\bf{3}}}} \right)}&{{\bf{for0}} \le {{\bf{x}}_{\bf{i}}} \le {\bf{1}}\,\,\left( {{\bf{i = 1,2,3}}} \right)}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\)
Determine\(\left( {\bf{a}} \right)\)the value of the constant c;
\(\left( {\bf{b}} \right)\)the marginal joint p.d.f. of\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{3}}}\); and
\(\left( {\bf{c}} \right)\)\({\bf{Pr}}\left( {{{\bf{X}}_{\bf{3}}}{\bf{ < }}\frac{{\bf{1}}}{{\bf{2}}}\left| {{{\bf{X}}_{\bf{1}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{ = }}\frac{{\bf{3}}}{{\bf{4}}}} \right.} \right){\bf{.}}\)
Suppose that\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{2}}}\)are i.i.d. random variables, and that each has the uniform distribution on the interval[0,1]. Evaluate\({\bf{P}}\left( {{{\bf{X}}_{\bf{1}}}^{\bf{2}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}^{\bf{2}} \le {\bf{1}}} \right)\)
Question:Suppose that in a certain drug the concentration of aparticular chemical is a random variable with a continuousdistribution for which the p.d.f.gis as follows:
\({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{8}}}{{\bf{x}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)
Suppose that the concentrationsXandYof the chemicalin two separate batches of the drug are independent randomvariables for each of which the p.d.f. isg. Determine
(a) the joint p.d.f.of X andY;
(b) Pr(X=Y);
(c) Pr(X >Y );
(d) Pr(X+Y鈮�1).
Consider the situation described in Example 3.7.14. Suppose that \(\) \({X_1} = 5\) and\({X_2} = 7\)are observed.
a. Compute the conditional p.d.f. of \({X_3}\) given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\).
b. Find the conditional probability that \({X_3} > 3\)given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\)and compare it to the value of \(P\left( {{X_3} > 3} \right)\)found in Example 3.7.9. Can you suggest a reason why the conditional probability should be higher than the marginal probability?
Suppose that either of two instruments might be used for making a certain measurement. Instrument 1 yields a measurement whose p.d.f.\({{\bf{h}}_{\bf{1}}}\)is
\({{\bf{h}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{2x}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)
Instrument 2 yields a measurement whose p.d.f.\({{\bf{h}}_2}\)is
\({{\bf{h}}_{\bf{2}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{3}}{{\bf{x}}^{\bf{2}}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)
Suppose that one of the two instruments is chosen randomly, and a measurement X is made with it.
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