Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

Show that $Y = a + b X$ , then
$ρ (X, Y) = \{\begin{cases} + 1 if b > 0 \\ - 1 if b < 0 \end{cases}$

We prove that

$Y = a + b X$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

Given in the question that, We have to find

Show that $Y = a + b X$ .

Step 2: Explanation

If $Y = a + b X$ with parameters $a$ and $b$

$ρ (X, Y) = \frac{Cov (X, Y)}{\sqrt{Var (X)} \sqrt{Var (Y)}}$

$Cov (X, Y) = Cov (X, a + b X)$

$Var (Y) = Var (a + b X)$

$= o + b^{2} Var (X)$

Now,

$ρ (X, Y) = \frac{b Var (X)}{\sqrt{Var (X)} \sqrt{b^{2} Var (X)}}$

Finally,

$ρ (X, Y) = \{\begin{cases} + 1 if b > 0 \\ - 1 if b < 0 \end{cases}$

Step 3: Final answer

We prove that

$Y = a + b X$

and

$ρ (X, Y) = \{\begin{cases} + 1 if b > 0 \\ - 1 if b < 0 \end{cases}$

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A First Course in Probability

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Short Answer

Show that $Y = a + b X$ , then
$ρ (X, Y) = \{\begin{cases} + 1 if b > 0 \\ - 1 if b < 0 \end{cases}$

Step by Step Solution

Step 1: Given information

Step 2: Explanation

Step 3: Final answer

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Show that $Y = a + b X$ , then
$ρ (X, Y) = \{\begin{cases} + 1 if b > 0 \\ - 1 if b < 0 \end{cases}$