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$X(t)$ is a stochastic process defined on the time interval $(0,T)$. Discretizing the time interval one can specify a random variable $X(t_i)$ as:

$$t_0= 0 < t_1,t_2,...,t_{n−1},t_n=T$$

And may be considered as being dependent on the previous random variables $X(t_1),X(t_2),...,X(t_{i−1})$ and $X(t_i)$ may be considered as independent on the random variables that follows in time $X(t_{i+1}),...,X(t_n)$? The conditional probabilities

$$ P(X(t_i) < x|X(t_{i−1}))=P(X(t{i−1}),X(t_i) < x)P(X(t_{i−1})) $$

and

$$ P(X(t_i) < x|X(t_i+1)) = P(X(t_i) < x,X(t_{i+1}) < x)P(X(t_{i+1}) < x)=P(X(t_i) $$

they end up contradicting each other since one has taken

$$P(X(t_i)<x|X(t_{i+1}))=P(X(t_i)<x)$$

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    $\begingroup$ Proper use of subscripts using LaTeX might help readability. $\endgroup$
    – Henry
    Commented Jan 27 at 0:48
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    $\begingroup$ Independence has a particular meaning in statistics and is not directly related to causality. In particular, "$X_1$ is independent of $X_2$ but $X_2$ is dependent on $X_1$" is not a correct statement. If knowing the value of $X_{t_i}$ provides information about the distribution of $X_{t_{i+1}}$ then knowing the value of $X_{t_{i+1}}$ provides information about the distribution of $X_{t_{i}}$ $\endgroup$
    – Henry
    Commented Jan 27 at 0:51
  • $\begingroup$ Hi. Welcome to CV. This is a MathJax-enabled site. Please use this facility to typeset your mathematical expressions for better legibility. For a quick guide, check this Meta CV post: Instructions on how to use LaTeX on CrossValidated. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Feb 15 at 14:14
  • $\begingroup$ I have attempted to reformat your code using LaTeX/MathJax, If there is a mistake please roll it back, but hopefully my edit(s) will make the question more readable. Well, I thought I had edited it, but I guess it needs to be reviewed by a moderation person. $\endgroup$
    – user1029384756
    Commented Feb 17 at 17:04
  • $\begingroup$ I tried to clean up this question, but the wording here is not always super clear, I would try to make the question a little more readable if you want an answer from somebody here. $\endgroup$
    – Shawn Hemelstrand
    Commented Feb 17 at 23:36

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Two random variables X1 and X2 may be partially dependent i.e. X1 is independent of X2 but X2 is dependent on X1?

No. Statistical dependence/independence is always a two-way relation, because it is not about causality (as a comment mentioned).

Consider for a specific region, $X$="number of sandstorms per month" and $Y$ ="carwash revenues per month". We could reasonably argue that $X$ affects $Y$, co-determines $Y$, and so there exists a causal link from $X$ to $Y$. This is one-way, since we do not expect that this $Y$ affects this $X$.

But once there exists such a one-way causal link, statistical dependence arises as a two-way relation: if I know something about the probabilities governing $X$, "I can say something" about the probabilities governing $Y$; and if I know something about the probabilities governing $Y$, I can say something about the probabilities governing $X$.

This is one of the main reasons why "correlation does not establish causation" (the other being the case where correlation between two variables is caused by the existence of a third variable that affects causally both).

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