Chapter 7: Q.23 (page 360)

Show that $Z$ is a standard normal random variable and if $Y$ is defined by $Y = a + b Z + c Z^{2}$ , then
$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$

Short Answer

Expert verified

We prove that $Y = a + b Z + c Z^{2}$

Step by step solution

Given information

Given in the question that, $Y = a + b Z + c Z^{2}$ .

Explanation

Let $Z$ is a standard normal random variable $N (0, 1)$ , and $Y = a + b Z + c Z^{2}$

Now, it can be shown $ρ (Y, Z)$

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

$= b Var (Z) + c E (Z^{3}) - c E (Z^{2}) E (Z)$

$E (Z) = 0, Var (Z) = 1$ , because $Z$ are standard normal random variable.

$E (Z^{3}) = 0$ order moments are always zero,

$Cov (Y, Z) = b Var (Z) = b$

$Var (Y) = b^{2} Var (Z) + c^{2} Var (Z^{2})$

$= b^{2} + c^{2} E (Z^{4}) - c^{2} E {(Z^{2})}^{2}$

$= b^{2} + 3 c^{2} - 1 c^{2}$

$= b^{2} + 2 c^{2}$

$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}} \sqrt{1}} = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$

Hence, we obtain:

$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$

Final answer

We prove that $Y = a + b Z + c Z^{2}$

and

$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$

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Show that $Z$ is a standard normal random variable and if $Y$ is defined by $Y = a + b Z + c Z^{2}$ , then
$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$

Short Answer

Step by step solution

Given information

Explanation

Final answer

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Show that Zis a standard normal random variable and if Yis defined by Y=a+bZ+cZ2, thenρ(Y,Z)=bb2+2c2

Short Answer

Step by step solution

Given information

Explanation

Final answer

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Most popular questions from this chapter

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Show that $Z$ is a standard normal random variable and if $Y$ is defined by $Y = a + b Z + c Z^{2}$ , then
$ρ (Y, Z) = \frac{b}{\sqrt{b^{2} + 2 c^{2}}}$