Please show the full steps. Thanks! Let (X,Y) be BVN(0,0, 1, 1, p). Find the variance of XY.

Name: Please show the full steps. Thanks! Let (X,Y) be BVN(0,0, 1, 1, p). Fi
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Description: Please show the full steps. Thanks! Let (X,Y) be BVN(0,0, 1, 1, p). Find the variance of XY.

Given that; (X, Y)~BVN(0, 0, 1, 1, ρ)⇒X~Normal(0,1) and Y~Normal(0,1)Y|X=x~Normal(ρx, 1-ρ2) We know;…

Answered: Let (X, Y) be BVN(0,0, 1, 1, p). Find…

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Please show the full steps. Thanks!

Let (X,Y) be BVN(0,0, 1, 1, p). Find the variance of XY.

Expert Solution

Step 1

Given that;

$(X, Y) ~ B V N (0, 0, 1, 1, ρ) \Rightarrow X ~ N o r m a l (0, 1) a n d Y ~ N o r m a l (0, 1) Y | X = x ~ N o r m a l (ρ x, 1 - ρ^{2})$

We know;

$E (X^{2}) = 1, E (Y^{2}) = 1 E (X^{4}) = 3, E (Y^{4}) = 3 a n d; E (X Y) = ρ$

Now;

$V a r (Y | X = x) = 1 - ρ^{2} \Rightarrow E (Y^{2} | X = x) - E^{2} (Y | X = x) = 1 - ρ^{2} \Rightarrow E (Y^{2} | X = x) - {(ρ x)}^{2} = 1 - ρ^{2} \Rightarrow E (Y^{2} | X = x) = {(ρ x)}^{2} + 1 - ρ^{2}$

Hence;

$E (X^{2} Y^{2}) = E \{E (X^{2} Y^{2} | X)\} = E \{X^{2} E (Y^{2} | X)\} = E \{X^{2} (ρ^{2} X^{2} + 1 - ρ^{2})\} = ρ^{2} E (X^{4}) + (1 - ρ^{2}) E (X^{2}) = 3 ρ^{2} + 1 - ρ^{2} = 2 ρ^{2} + 1$

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Let (X, Y) be BVN(0,0, 1, 1, p). Find the variance of XY.