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I found this notation in one paper that focuses on copulas:

Consider a $d$ -dimensional continuous random vector $X=\left\{X_{1}, X_{2}, \cdots X_{d}\right\}$ with marginals $F_{i}\left(x_{i}\right)=$ $\mathbb{P}\left(X_{i} \leq x_{i}\right) .$ Given a $x \in \mathbb{R}^{d},$ the distribution function $F(x)=\mathbb{P}\left(X_{1} \leq x_{1}, \cdots X_{d} \leq x_{d}\right)$ specifies all marginal distributions $F_{i}\left(x_{i}\right)$ as well as any dependencies between $X$.

I understood $X_i$ are random variables. I would kindly ask what is the $x_i$ in this case. It should also be random variable, but then there is just $x$ which is a random vector then.

When I think of marginal probabilities I imagine something like this:

  $X_1$ $X_2$ M
  10 20 30
  30 10 40
  20 10 30
Totals 60 40 100

So marginal probabilities are these form the M column. So isn't it better to define the marginal probabilities with the uppercase letter, just for consistency. Your sanity may help me understand the notations.

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The vector $X=(X_1,\ldots,X_n)$ is a random vector made of random variables as components. For instance, $X=(X_1,X_2)$ with $X_1\sim\mathcal N(0,1)$ and $X_2|X_1\sim\mathcal N(-2X_1,2^2)$ is such a thing. The marginal distribution of $X_1$ is $\mathcal N(0,1)$ and the marginal distribution of $X_2$ is $\mathcal N(0,8)$

The joint cdf is defined as $$F(x)=F(x_1,x_2,\ldots,x_n)=\mathbb P(X_1\le x_1,\cdots,X_n\le x_n)$$ where $x=(x_1,\ldots,x_n)\in\mathbb R^n$. In the above example, $$F(x)=F(x_1,x_2)=\mathbb P(X_1\le x_1,X_2\le x_2)=\int_{-\infty}^{x_1} \varphi(x_1)\Phi([x_2+2x_1]/2)\,\text dx_1$$ It defines marginal distributions${}^*$ through their cdfs $$F_i(x_i)=F(+\infty,\ldots,\underbrace{+\infty,x_i,+\infty,}_\text{$i$-th position}\ldots,+\infty)$$


${}^*$I do not understand the table produced in the question, as the probabilities (?) do not add up to 100.

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  • $\begingroup$ Are the $x_i$ scalars then? $\endgroup$
    – Good Luck
    Mar 12, 2021 at 14:33
  • $\begingroup$ I will gladly accept the answer if you touch the later casing part. $\endgroup$
    – Good Luck
    Mar 12, 2021 at 14:44
  • $\begingroup$ Am I wrong when I imagine $x_i$ is a sum of $i$-th state of random variables $X_i$? You see my upper table and just replace $M$ with $x$ then. $\endgroup$
    – Good Luck
    Mar 12, 2021 at 14:47
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    $\begingroup$ The table is not appropriate because the rows would correspond to the possible values of the rv $X_1$ while the columns would correspond to the possible values of the rv $X_2$. $\endgroup$
    – Xi'an
    Mar 12, 2021 at 15:09
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    $\begingroup$ Gender would be the random variable $X_1$ with possible values "female" and "male" while the number of ice-creams per day would be the random variable $X_2$. These two random variables have a joint distribution given by a contingency table. $\endgroup$
    – Xi'an
    Mar 12, 2021 at 15:34

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