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Chapter 5: Q10SE (page 344)

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.

Short Answer

Expert verified

\({Y_1}{\rm{ and }}{Y_2}\) have a bivariate normal distribution.

Step by step solution

01

Given information and the pdf

Let \(X = {\left| {{X_1}\,\,{X_2}} \right|^T}\). Since \({X_1}\,,{X_2}\)has a bivariate normal distribution, therefore the pdf is:

\({f_X}\left( x \right) = \frac{1}{{2\pi \sqrt {\det \left( \Sigma \right)} }}\exp \left\{ {\frac{{ - 1}}{{2\left( {1 - {p^2}} \right)}}\left[ {{{\left( {x - \mu } \right)}^T}{\Sigma ^{ - 1}}\left( {x - \mu } \right)} \right]} \right\}\)

Where,

\(\begin{array}{l}\Sigma :2 \times 2\,\,{\rm{matrix}}\\\Sigma \mu :2 \times 1\,\,{\rm{vector}}\end{array}\)

02

Define the variable Y and find the Jacobian

Let \(Y = {\left| {{Y_1}\,\,{Y_2}} \right|^T}\). The Jacobian matrix is

\(\begin{array}{c}J = \left( {\begin{array}{*{20}{c}}{\frac{{\partial {y_1}}}{{\partial {x_1}}}}&{\frac{{\partial {y_1}}}{{\partial {x_2}}}}\\{\frac{{\partial {y_2}}}{{\partial {x_1}}}}&{\frac{{\partial {y_2}}}{{\partial {x_2}}}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}\\{{a_{21}}}&{{a_{22}}}\end{array}} \right)\end{array}\)

Now, by condition, \(\left| {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}\\{{a_{21}}}&{{a_{22}}}\end{array}} \right| \ne 0\)\(\)

This further helps in solving \(X = {J^{ - 1}}\left( {Y - b} \right),\,\,b = {\left[ {{b_1}\,\,{b_2}} \right]^T}\)

03

Find the pdf of Y

\(\begin{aligned}{}{f_Y}\left( y \right) &= {f_X}\left( {{J^{ - 1}}\left( {y - b} \right)} \right)\frac{1}{{\left| {\det \left( J \right)} \right|}}\\ &= \frac{1}{{2\pi \sqrt {\det \left( \Sigma \right)\left( {\det {{\left( J \right)}^2}} \right)} }}\exp \left[ { - \frac{1}{2}{{\left( {y - b - J\mu } \right)}^T}{{\left( {{J^T}} \right)}^{ - 1}}{\Sigma ^{ - 1}}{J^{ - 1}}\left( {y - b - J\mu } \right)} \right]\\ &= \frac{1}{{2\pi \sqrt {\left( {\det \left( {J\Sigma {J^T}} \right)} \right)} }}\exp \left[ { - \frac{1}{2}{{\left( {y - b - J\mu } \right)}^T}{{\left( {J{\Sigma ^{ - 1}}{J^T}} \right)}^{ - 1}}\left( {y - b - J\mu } \right)} \right] \ldots \left( 1 \right)\end{aligned}\)

04

Use the following transformation rules

The proof has been used by following transformation

\(\begin{aligned}{}\det \left( {{A^T}} \right) &= \det \left( A \right)\\\det \left( {AB} \right) &= \det \left( A \right)\det \left( B \right)\\{\left( {AB} \right)^{ - 1}} &= {B^{ - 1}}{A^{ - 1}}\end{aligned}\)

According to (1), \({Y_1}{\rm{ }}and{\rm{ }}{Y_2}\) have a bivariate normal distribution.

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Most popular questions from this chapter

Let\({X_1},{X_2},{X_3}\)be a random sample from the exponential distribution with parameter\(\beta \). Find the probability that at least one of the random variables is greater than t, where\(t > 0\)

Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having the normal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, we shall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).

a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distribution with both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).

Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).

Now show that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normal and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .

b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).

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

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\({X_1} - 3{X_2}\).

Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having thenormal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, weshall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).

a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distributionwith both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).

Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).

Nowshow that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normals and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .

b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).
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