Riemann surface
matlab

Exercise 1.1 Given a function $g$ on $\mathbb{R}$, state the two basic conditions for $g$ to be a probability density function (pdf) with respect to the Lebesgue measure. Recall the definition of the cumulative distribution function (cdf) associated with $g$ and that of the quantile function of $g$..

If $g$ is integrable with respect to the Lebesgue measure, $g$ is a pdf if and only if

$g$ is non-negative, $g(x) \geq 0$

$g$ integrates to 1 ,
$$\int_{\mathbb{R}} g(x) \mathrm{d} x=1$$

Exercise 1.2 If $\left(x_1, x_2\right)$ is a normal $\mathcal{N}_2\left(\left(\mu_1, \mu_2\right), \Sigma\right)$ random vector, with
$$\Sigma=\left(\begin{array}{cc} \sigma^2 & \omega \sigma \tau \ \omega \sigma \tau & \tau^2 \end{array}\right),$$
recall the conditions on $(\omega, \sigma, \tau)$ for $\Sigma$ to be a (nonsingular) covariance matrix. Under those conditions, derive the conditional distribution of $x_2$ given $x_1$.

The matrix $\Sigma$ is a covariance matrix if

$\Sigma$ is symmetric and this is the case;

$\Sigma$ is semi-definite positive, i.e. , for every $\mathbf{x} \in \mathbb{R}^2, \mathbf{x}^{\top} \Sigma \mathbf{x} \geq 0$, or, for every $\left(x_1, x_2\right)$,
$$\sigma^2 x_1^2+2 \omega \sigma \tau x_1 x_2+\tau^2 x_2^2=\left(\sigma x_1+\omega \tau x_2\right)^2+\tau^2 x_2^2\left(1-\omega^2\right) \geq 0 .$$

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