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Chapter 3: Q15E (page 167)
Let X1鈥�,Xn be independent random variables, and let W be a random variable such that \({\rm P}\left( {w = c} \right) = 1\) for some constant c. Prove that \({x_1},....,{x_n}\)they are conditionally independent given W = c.
\({x_1}....{x_n}\)are conditionally independent given W=c
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LetXbe a random variable with a continuous distribution.
Let \({{\bf{x}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{2}}}{\bf{ = x}}\)
a.Prove that both \({{\bf{x}}_{\bf{1}}}\) and \({{\bf{x}}_{\bf{2}}}\) have a continuous distribution.
b.Prove that \({\bf{x = (}}{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{)}}\)does not have a continuous
joint distribution.
Suppose that the p.d.f. of a random variable X is as
follows:\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{2}x\,\,\,\,\,\,\,\,for\,0 < x < 2\\0\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
Also, suppose that \(Y = X\left( {2 - X} \right)\) Determine the cdf and the pdf of Y .
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).
Two students,AandB,are both registered for a certain course. Assume that studentAattends class 80 percent of the time, studentBattends class 60 percent of the time, and the absences of the two students are independent. Consider the conditions of Exercise 7 of Sec. 2.2 again. If exactly one of the two students,AandB,is in class on a given day, what is the probability that it isA?
Start with the joint distribution of treatment group and response in Table 3.6 on page 138. For each treatment group, compute the conditional distribution of response given the treatment group. Do they appear to be very similar or quite different?
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