Q7.23 Let Xi,Yi,i=1,鈥�, be a sequence... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q7.23 (page 365)

Let $(X_{i}, Y_{i}), i = 1, 鈥�$ , be a sequence of independent and identically distributed random vectors. That is, $X_{1}, Y_{1}$ is independent of, and has the same distribution as, $X_{2}, Y_{2}$ , and so on. Although $X_{i}$ and $Y_{i}$ can be dependent, $X_{i}$ and $Y_{j}$ are independent when $i 鈮� j$ . Let
$渭_{x} = E [X_{i}], 渭_{y} = E [Y_{i}], 蟽_{x}^{2} = Var (X_{i})$
$蟽_{y}^{2} = Var (Y_{i}), 蚁 = Corr (X_{i}, Y_{i})$

Find $Corr ({鈭�}_{i = 1}^{n} X_{i}, {鈭�}_{j = 1}^{n} Y_{j})$ .

Short Answer

Expert verified

Therefore,
$Corr (\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j}) = 蚁$

Step by step solution

Concept Introduction

Let $(X_{i}, Y_{i})$ for all $i = 1, 2, 鈥�$ be a sequence of independent and identically distributed random vectors.

Explanation

Let $(X_{i}, Y_{i})$ for all $i = 1, 2, 鈥�$ be a sequence of independent and identically distributed random vectors.
From the given information:

$渭_{x} = E (X_{i})$ and $渭_{y} = E (Y_{i})$

$蟽_{x}^{2} = Var (X_{i})$ and $蟽_{y}^{2} = Var (Y_{i})$

Explanation

$蚁 = Corr (X_{i}, Y_{i})$

The objective is to find Corr $(\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j})$ .

Explanation

From the Correlation properties:
$Corr (\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j}) = \frac{Cov ({鈭�}_{i} X_{i}, {鈭�}_{j} Y_{j})}{{(Var ({鈭�}_{i} X_{i}) Var ({鈭�}_{j} Y_{j}))}^{\frac{1}{2}}}$

$= \frac{{鈭�}_{i} {鈭�}_{j} Cov (X_{i}, Y_{j})}{{(n 路蟽_{x}^{2} 路 n 路蟽_{y}^{2})}^{\frac{1}{2}}}$

Explanation

Simplifying the expression and using the fact that $Cov (X_{i}, Y_{i}) = 蚁路蟽_{x} 路蟽_{y}$ :

$Corr (\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j}) = \frac{{鈭�}_{i} Cov (X_{i}, Y_{i}) + {鈭�}_{i} {鈭�}_{j = i} Cov (X_{i}, Y_{i})}{n 路蟽_{x} 路蟽_{y}}$

$= \frac{n 路蚁路蟽_{x} 路蟽_{y}}{n 路蟽_{x} 路蟽_{y}}$

$= 蚁$

Therefore,
$Corr (\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j}) = 蚁$

Step6:Final Answer

$Corr (\underset{i}{鈭�} X_{i}, \underset{j}{鈭�} Y_{j}) = 蚁$

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Short Answer

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Concept Introduction

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Step6:Final Answer

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