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### Conditional distribution for 3 variables [duplicate]

Everywhere it is done for bivariate or given a hints. But Not in details for trivariate.
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### How to find mean of multivariate normal distribution when holding a variable constant? [duplicate]

I wanted to know if there is a way to calculate the mean of a multivariate normal distribution when a certain variable is held constant. For example, if I had a continuous bivariate normal ...
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### Conditional mean independence implies unbiasedness and consistency of the OLS estimator

Consider the following multiple regression model: $$Y=X\beta+Z\delta+U.\tag{1}$$ Here $Y$ is a $n\times 1$ column vector; $X$ a $n\times (k+1)$ matrix; $\beta$ a $(k+1)\times 1$ column vector; $Z$ a ...
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### Conceptual proof that conditional of a multivariate Gaussian is multivariate Gaussian

I understand the arithmetic derivation of the PDF of a conditional distribution of a multivariate Gaussian, as explained here, for example. Does anyone know of a more conceptual (perhaps, co-ordinate ...
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### Expressing conditional covariance matrix in terms of covariance matrix

Suppose we have two multivariate random variables $\mathbf{X}$ (of dimension $n_x$) and $\mathbf{Y}$ (of dimension $n_y$). The covariance matrix $C_{X,Y}$ can be written as the following block-matrix ...
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### What is the probability that $X<Y$ given $\min(X,Y)$?

Suppose $X$ and $Y$ are bivariate normal with mean $\mu=(\mu_1,\mu_2)$ and covariance \$\Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \\ \...