Paper 2, Section II, F

Let $X_{1}, X_{2}$ be bivariate normal random variables, with the joint probability density function

$f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right)=\frac{1}{2 \pi \sigma_{1} \sigma_{2} \sqrt{1-\rho^{2}}} \exp \left[-\frac{\varphi\left(x_{1}, x_{2}\right)}{2\left(1-\rho^{2}\right)}\right]$

where

$\varphi\left(x_{1}, x_{2}\right)=\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2 \rho\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}$

and $x_{1}, x_{2} \in \mathbb{R}$.

(a) Deduce that the marginal probability density function

$f_{X_{1}}\left(x_{1}\right)=\frac{1}{\sqrt{2 \pi} \sigma_{1}} \exp \left[-\frac{\left(x_{1}-\mu_{1}\right)^{2}}{2 \sigma_{1}^{2}}\right]$

(b) Write down the moment-generating function of $X_{2}$ in terms of $\mu_{2}$ and $\sigma_{2} \cdot[N o$ proofs are required.]

(c) By considering the ratio $f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right) / f_{X_{2}}\left(x_{2}\right)$ prove that, conditional on $X_{2}=x_{2}$, the distribution of $X_{1}$ is normal, with mean and variance $\mu_{1}+\rho \sigma_{1}\left(x_{2}-\mu_{2}\right) / \sigma_{2}$ and $\sigma_{1}^{2}\left(1-\rho^{2}\right)$, respectively.