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2

If you have two independent multivariate normal distributions then their joint distribution is another multivariate normal distribution with a block scheme covariance matrix $$\begin{bmatrix} \Sigma_1&0\\0&\Sigma_2 \end{bmatrix}$$ When you also consider the joint distribution with a Wishart distribution then you get a Normal-Wishart distribution


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A process $X_t$ is WSS when ($K_{XX}$ is an autocovariance function): $$ \forall\tau\in\mathbb{R}: \mathbb{E}[X_t]=\mathbb{E}[X_{t+\tau}] $$ $$ \forall t_1,t_2\in\mathbb{R}:K_{XX}(t_1, t_2)=K_{XX}(t_1-t_2,0) $$ $$ \forall t\in\mathbb{R}:\mathbb{E}[|X_t|^2] < \infty $$ You can note, that this definition is symmetric with respect to substitution $t\...


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$\Theta \sim U(-\pi, \pi)$ so the density of $\Theta$ is given by $\frac{1}{2 \pi}$ in $-\pi, \pi$. $F(x, y) = P(X \le x, Y \le y) = P(\Theta \le \arcsin(x) \wedge \arccos(y))$. $ F(x, y) = \frac{1}{2\pi}\int_{-\pi}^{\arcsin(x) \wedge \arccos(y)} 1 \cdot d\theta = \frac{1}{2\pi}\arcsin(x) \wedge \arccos(y) + \frac{1}{2}. $


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The support of the random vector $(X_j, Y)$ is the set $\mathcal A = \left \{ (p,q) \in \mathbb N^2 \mid p \leq q \right \}$. The distribution of $(X_j, Y)$ is given by the probabilities this vector takes for each elements of $\mathcal A$. Thus let $p \leq q$, then: \begin{align*} \mathbb P\left (X_j = p, Y= q \right ) &= \mathbb P \left (Y=q \mid X_j = ...


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For discrete random variables the raking/iterative proportional fitting algorithm constructs a joint distribution if one exists (under some additional assumption about zero cells). It works for marginal distributions of any order, and not necessarily the same order for each margin. IPF is (or was) used to fit loglinear models -- its guaranteed convergence ...


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