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Chapter 3: Q 13E (page 152)

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?

Short Answer

Expert verified

The fourth one looks quite different from the others, especially from the second.

Step by step solution

01

Given information

Referring to the Table 3.6.

02

Necessary calculation

There are four different treatments on which conditional distribution.

The marginal p.d.f. of treatment Y is given in the bottom row of the referring table in the text.

The conditional p.d.f. of the response given each treatment is the ratio of the two rows above that to the bottom row.

\({g_1}\left( {x\left| 1 \right.} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{0.120}}{{0.267}} = 0.449438}&{if\,x = 0}\\{\frac{{0.147}}{{0.267}} = 0.550562}&{if\,x = 1}\end{array}} \right.\)

\({g_1}\left( {x\left| 2 \right.} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{0.087}}{{0.253}} = 0.343874}&{if\,x = 0}\\{\frac{{0.166}}{{0.253}} = 0.656126}&{if\,x = 1}\end{array}} \right.\)

\({g_1}\left( {x\left| 3 \right.} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{0.146}}{{0.253}} = 0.577075}&{if\,x = 0}\\{\frac{{0.107}}{{0.253}} = 0.422925}&{if\,x = 1}\end{array}} \right.\)

\({g_1}\left( {x\left| 4 \right.} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{0.160}}{{0.227}} = 0.704846}&{if\,x = 0}\\{\frac{{0.067}}{{0.227}} = 0.295154}&{if\,x = 1}\end{array}} \right.\)

Therefore, the fourth one looks quite different from the others, especially from the second.

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Most popular questions from this chapter

Find the unique stationary distribution for the Markov chain in Exercise 2.

Suppose that each of two gamblersAandBhas an initial fortune of 50 dollars and that there is a probabilitypthat gamblerAwill win on any single play of a game against gamblerB. Also, suppose either that one gambler can win one dollar from the other on each play of the game or that they can double the stakes and one can win two dollars from the other on each play of the game. Under which of these two conditions doesAhave the greater

probability of winning the initial fortune ofBbefore losing her own for each of the following conditions: (a)\(p < \frac{1}{2}\);

(b)\(p > \frac{1}{2}\); (c)\(p = \frac{1}{2}\)?

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

\(\left( {\bf{c}} \right)\)\({\bf{Pr}}\left( {{{\bf{X}}_{\bf{3}}}{\bf{ < }}\frac{{\bf{1}}}{{\bf{2}}}\left| {{{\bf{X}}_{\bf{1}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{ = }}\frac{{\bf{3}}}{{\bf{4}}}} \right.} \right){\bf{.}}\)

Question:Suppose thatXandYhave a discrete joint distributionfor which the joint p.f. is defined as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{30}}}}\left( {{\bf{x + y}}} \right)\;{\bf{for}}\;{\bf{x = 0,1,2}}\;{\bf{and}}\;{\bf{y = 0,1,2,3}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

a. Determine the marginal p.f.鈥檚 ofXandY.

b. AreXandYindependent?

Question: Suppose that the joint p.d.f. of two random variablesXandYis as follows:

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}c\left( {{x^2} + y} \right)\,\,\,\,for\,0 \le y \le 1 - {x^2}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)

Determine (a) the value of the constantc;

\(\begin{array}{l}\left( {\bf{b}} \right)\,{\bf{Pr}}\left( {{\bf{0}} \le {\bf{X}} \le {\bf{1/2}}} \right){\bf{;}}\,\left( {\bf{c}} \right)\,{\bf{Pr}}\left( {{\bf{Y}} \le {\bf{X + 1}}} \right)\\\left( {\bf{d}} \right)\,{\bf{Pr}}\left( {{\bf{Y = }}{{\bf{X}}^{\bf{2}}}} \right)\end{array}\)
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