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Statistical Notations confuses me a lot and I get lost easily in following when the authors are talking about random variables vs observations, probabilities, probability density functions or distributions etc.

Anyway in Pearl's "Causal Inference in Statistics: An Overview" (2009) I understood the following DAG to depict the causal relationship between Random Variables Z, X and Y. enter image description here

but the corresponding non parametric functional equations are using small lower case letters:

$$ z = f_Z(u_Z) $$ $$ x = f_X(z,u_X) $$ $$ y = f_Y(x,u_Y) $$

I interpret this that the lowercase letter represent an observation of the Random Variables. Is that assumption correct? How am I supposed to interpret the subscript for $f$ and $u$?

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Yes, you are correct about upper-case versus lower-case: the upper-case $U_X,$ for example, is the exogenous random variable. Any particular value of that random variable is $u_X.$ This is a fairly common statistical notation practice: the upper-case is the random variable, the lower-case a particular value of that random variable.

The subscripts for $f$ and $u$ are there simply to distinguish them: it is not the case that the form of the functions for $x, y,$ and $z$ are all the same, nor is it the case that the exogenous variables $U_X, U_Y,$ and $U_Z$ are all identical; far from it, we usually assume the exogenous variables are independent, though there are ways to handle dependence.

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  • $\begingroup$ Thank you for clarifying! I have a few more questions, I hope this is ok. Can I similarly assume that $x$ is then an observation from the random variable X. Which is drawn from a distribution, perhaps P(x,y,z) or P(x|z) ? How come the author uses capital P to denote such distributions and not lower p? And when Pearl says „changes to Z“ can I think of it in such a way that the related distribution is changed? $\endgroup$
    – amsulic
    Feb 8, 2023 at 21:14
  • $\begingroup$ Yes, $x$ is an actual value of the random variable $X.$ $X$ can be drawn from a variety of distributions, it is true; not sure if something like $P(X,Y,Z)$ would be allowed - that's a joint distribution. You'd have to take a sample $(X,Y,Z),$ I think, from such a distribution. You could draw from $P(X|Z).$ Capital $P$ versus lowercase $p$ is not usually a terribly important distinction in my experience. Need more context to discuss the "changes to $Z$" comment. $\endgroup$ Feb 8, 2023 at 21:23

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