Why is the conditional variance of Y equal to (1-rou^2)* variance of y?

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The discussion focuses on understanding the conditional variance of Y in relation to the multivariate normal distribution. It highlights that the conditional variance can be derived from the marginal distribution of X and the conditional probability density function (pdf) of Y given X. The equation for the conditional variance is expressed as (1-rou^2) times the variance of Y. Additionally, the joint distribution of two independent standard normal random variables is confirmed to be bivariate normal through the use of the Jacobian. The conversation emphasizes the importance of deriving the appropriate distributions to understand these relationships clearly.
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You have to first derive the marginal distribution of X. Then the conditional pdf of Y|X is given by f(x,y)/f(x). It will follow from there that the conditional variance is (1-p^2)Var(Y).

I am having problems with this question myself:

Let X1 and X2 be independent standard normal random variables. Show that the joint distribution of

Y1=aX1 + bX2 + c
Y2=dX1 + eX2 + f

is bivariate normal.
 
Your one can be shown using the jacobian

J=|a b| =ae-bd
d e

then the joint pdf of y1 and y2= joing pdf of X1 and X2 / (ae-bd)
X1 and X2 are independent, you can easily get their joint pdf,
and you can get joint pdf of y1 and y2, it is also a joint normal
ych22 said:
You have to first derive the marginal distribution of X. Then the conditional pdf of Y|X is given by f(x,y)/f(x). It will follow from there that the conditional variance is (1-p^2)Var(Y).

I am having problems with this question myself:

Let X1 and X2 be independent standard normal random variables. Show that the joint distribution of

Y1=aX1 + bX2 + c
Y2=dX1 + eX2 + f

is bivariate normal.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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