Suppose I have 3 standard normal random variables: $X$, $Y$, and $Z$. Suppose I know the correlation between $(X,Y)$, $(X,Z)$, and $(Y,Z)$ but I do not have access to the raw data.
Now I want to perform multivariable linear regression, predicting $Z$ from $X$ and $Y$:
$$Z = a_0 + a_1 \cdot X + a_2 \cdot Y + \epsilon$$
Is this possible to do? If not what additional information would make it possible (besides the raw data)
$$X=\begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} \sim \mathcal{N}_3 (\mu,\Sigma) \ \ \ \ \ \ \mu=\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix} \ \ \ \ \ \ \Sigma=\begin{bmatrix} \sigma^2_1 & \sigma_{1,2} & \sigma_{1,3} \\ \sigma_{1,2} & \sigma^2_2 & \sigma_{2,3} \\ \sigma_{1,3} & \sigma_{2,3} & \sigma^2_3 \end{bmatrix}$$ We want to find the conditional density of $X_1|X_2=x_2,X_3=x_3$ $$f_{X_1|X_2,X_3}(x_1,x_2,x_3)=\frac{f_{X_1,X_2,X_3}(x_1,x_2,x_3)}{f_{X_2,X_3}(x_2,x_3)}$$ But we know that $$\begin{bmatrix} X_2 \\ X_3 \end{bmatrix} \sim \mathcal{N}_2 (\mu_{2,3},\Sigma_{2,3}) \ \ \ \ \ \ \mu_{2,3}=\begin{bmatrix} \mu_2 \\ \mu_3 \end{bmatrix} \ \ \ \ \ \ \Sigma_{2,3}=\begin{bmatrix} \sigma^2_2 & \sigma_{2,3} \\ \sigma_{2,3} & \sigma^2_3 \end{bmatrix}$$ So $$f_{X_1|X_2,X_3}(x_1,x_2,x_3)=\frac{(2\pi)^{\frac{-3}{2}}\det(\Sigma)^{-\frac{1}{2}}e^{-0.5(\mathbf{x}-\mu)^\textrm{T}\Sigma^{-1}(\mathbf{x}-\mu)}}{(2\pi)^{-1}\det(\Sigma_{2,3})^{-\frac{1}{2}}e^{-0.5(\mathbf{x}_{2,3}-\mu_{2,3})^\textrm{T}\Sigma_{2,3}^{-1}(\mathbf{x}_{2,3}-\mu_{2,3})}}$$ It can be shown that this can be simplified to $$f_{X_1|X_2,X_3}(x_1,x_2,x_3)=\frac{1}{\sqrt{2\pi\gamma^2}}e^{-\frac{(x_1-\theta)^2}{2\gamma^2}}$$ where $$\theta=\mu_1+\begin{bmatrix} \sigma_{1,2} & \sigma_{1,3} \end{bmatrix}\begin{bmatrix} \sigma^2_2 & \sigma_{2,3} \\ \sigma_{2,3} & \sigma^2_3 \end{bmatrix}^{-1}\bigg(\begin{bmatrix} x_2 \\ x_3 \end{bmatrix}-\begin{bmatrix} \mu_2 \\ \mu_3 \end{bmatrix}\bigg)$$ $$\gamma^2=\sigma_1^2-\begin{bmatrix} \sigma_{1,2} & \sigma_{1,3} \end{bmatrix}\begin{bmatrix} \sigma^2_2 & \sigma_{2,3} \\ \sigma_{2,3} & \sigma^2_3 \end{bmatrix}^{-1}\begin{bmatrix} \sigma_{1,2} \\ \sigma_{1,3} \end{bmatrix}$$ notice that $\theta$ is the conditional expectation of $X_1$ given $X_2$ and $X_3$
Just to be more clear, set $$\begin{bmatrix} \beta_{2} & \beta_{3}\end{bmatrix}=\begin{bmatrix} \sigma_{1,2} & \sigma_{1,3} \end{bmatrix}\begin{bmatrix} \sigma^2_2 & \sigma_{2,3} \\ \sigma_{2,3} & \sigma^2_3 \end{bmatrix}^{-1}$$ Then $$\theta = \mu_1 + \beta_2(x_2-\mu_2)+\beta_3(x_3-\mu_3)= \\ = \underbrace{\mu_1 - \beta_2\mu_2-\beta_3\mu_3}_{=\alpha}+\beta_2x_2+\beta_3x_3$$ Therefore $$X_1|X_2,X_3 \sim \mathcal{N}(\alpha+\beta_2x_2+\beta_3x_3,\gamma^2)$$