Derive the characteristic function of a real-valued random variable which is normally distributed with mean $\mu$ and variance $\sigma^{2}$. What does it mean to say that an $\mathbb{R}^{n}$-valued random variable has a multivariate Gaussian distribution? Prove that the distribution of such a random variable is determined by its $\left(\mathbb{R}^{n}\right.$-valued) mean and its covariance matrix.

Let $X$ and $Y$ be random variables defined on the same probability space such that $(X, Y)$ has a Gaussian distribution. Show that $X$ and $Y$ are independent if and only if $\operatorname{cov}(X, Y)=0$. Show that, even if they are not independent, one may always write $X=a Y+Z$ for some constant $a$ and some random variable $Z$ independent of $Y$.

[The inversion theorem for characteristic functions and standard results about independence may be assumed.]