# Confused by the notation in DAG depicting structural causal model and corresponding functional equations

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.

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 $$u$$?

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.
• 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? Feb 8, 2023 at 21:14
• 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. Feb 8, 2023 at 21:23