# Way to define marginal probabilities

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.

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.
• Are the $x_i$ scalars then? Mar 12, 2021 at 14:33
• 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. Mar 12, 2021 at 14:47
• 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 $X_2$. Mar 12, 2021 at 15:09
• 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. Mar 12, 2021 at 15:34