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I have $(X,Y_1)$ with joint bivariate normal distribution. Also, $Y_1$ is conditional on $X$. Therefore $\rho$ is non-zero. Suppose I have $Y_2$ that is also normally distributed and $\mu_{Y_2} = \mu_{Y_1}$ and $\sigma_{Y_2}=\sigma_{Y_1}$. What is the relationship between $\text{Var}(Y_1|X)$ and $\text{Var}(Y_2)$ ?

I've been generating samples from $Y_1|X$ and $Y_2$ and it seems that for large sample sizes, the difference $|\text{Var}(Y_1|X)$ - $\text{Var}(Y_2)$| converges to a particular (small) number. This disagrees with theoretical result that I got by calculating $\sigma_{Y_1|X} = \sigma_{Y_1}\sqrt{1-\rho^2}$.

What is the interpretation of that?

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    $\begingroup$ I must not being understanding something. You specified in the problem that $\sigma_{Y_2}=\sigma_{Y_1}$. So the relationship is just that they're the same? If this isn't the case then please provide some more detail as to why not. $\endgroup$ – Dason Nov 7 '11 at 20:06
  • $\begingroup$ Sorry, I wasnt clear, I'm interested in comparing samples from Y1|X and Y2. $\endgroup$ – Saideira Nov 7 '11 at 20:18
  • $\begingroup$ Could you describe your simulation experiment which disagrees with the theoretical result? Thanks. $\endgroup$ – Xi'an Nov 9 '11 at 6:31
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First, "$Y_1$ is conditional on $X$" has no specific probabilistic meaning. Do you mean $X$ and $Y_1$ are not independent?

Second, if $Y_1$ and $Y_2$ have the same marginal normal distribution, then $$ \text{var}(Y_1|X) \le \text{var}(Y_1)=\text{var}(Y_2)\,. $$ More exactly, in general, $$ \text{var}(Y_1)=\text{var}(Y_2) = \text{var}(\mathbb{E}[Y_1|X])+\mathbb{E}[\text{var}(Y_1|X)]\,, $$ which means here that $$ \text{var}(Y_1)= \sigma_{Y_1}^2 = \rho^2\dfrac{\sigma_{Y_1}^2}{\sigma_X^2} \sigma_X^2+(1-\rho^2)\sigma_{Y_1}^2\,. $$ Third, I do not understand your simulation experiment: if you generate from the distribution of $Y_1$ conditional on $X$, this means you generate from a $$ \mathcal{N}\left(\rho \sigma_{Y_1} X / \sigma_X, \sqrt{1-\rho^2}\sigma_{Y_1}^2\right) $$ distribution (assuming $\mu_X=\mu_Y=0$). What am I missing?

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