Conditional distribution of trivariate normal

Question

Consider three non-independent normally distributed random variables $(T,S,C)$. I am interested in the distribution of $T$ conditional on $S=s$ and $C=c$.

I know that, for $\mu_T=\mu_S=\mu_C=0$ and $\sigma_T=\sigma_S=\sigma_C=1$, the conditional mean is given by

$E[T|S=s, C=c]=\beta_{TS;c} s+\beta_{TC;s} c$

where the betas are the regression coefficients: $\beta_{ij;k}=\frac{\rho_{ij}-\rho_{ik}\rho_{jk}}{\sqrt{1-\rho_{ik}^2}\sqrt{1-\rho_{jk}^2}}$

Is there a similar way to parameterise $Var[T|S=s, C=c]$?

NB. I'm not interested in a step-by-step derivation – I'd be equally happy with an expression derived from the symbolic integrals in Mathematica.

Answer 1

Assuming by "three non-independent normally distribution random variables" you mean that $(T,S,C)$ has a trivariate normal distribution with mean vector $(\mu_T,\mu_S,\mu_C)$ and covariance matrix

$$\Sigma=\left( \begin{array}{ccc} \sigma_T^2 & \rho_{TS} \sigma_S \sigma_T & \rho_{TC} \sigma_C \sigma_T \\ \rho_{TS} \sigma_S \sigma_T & \sigma_S^2 & \rho_{SC} \sigma_C \sigma_S \\ \rho_{TC} \sigma_C \sigma_T & \rho_{SC} \sigma_C \sigma_S & \sigma_C^2 \\ \end{array} \right)$$

Using Mathematica the conditional CDF for $T|S=s,C=c$ is

conditionalCDF = Probability[T <= t \[Conditioned] S == s && C == c, 
  {T, S, C} \[Distributed] MultinormalDistribution[μ, Σ]];

Then the conditional PDF is

conditionalPDF = D[conditionalCDF, t]

We see from inspection that the conditional pdf is that of a normally distributed random variable with mean

and variance

which can be simplified to

$$\mu_T+\frac{\sigma_T (\sigma_C (s-\mu_S) (\rho_{SC} \rho_{TC}- \rho_{TS})-\sigma_S (c-\mu_C) (\rho_{TC}-\rho_{SC} \rho_{TS}))} {\left(\rho_{SC}^2-1\right) \sigma_C \sigma_S}$$

and

$$\frac{\sigma_T^2 \left(\rho_{SC}^2-2 \rho_{SC} \rho_{TC} \rho_{TS}+\rho_{TC}^2+\rho_{TS}^2-1\right)}{\rho_{SC}^2-1}$$

respectively.