# Can we derive the distribution of a random variable based on a dependent random variable's distribution?

In the diagram below, there are three variables: X3 is a function of (depends on) X1 and X2, X2 also depends on X1. More specifically, X3 = f(X1, x2) and X2 = g(X1). Therefore, X3 = f(X1, g(X1)). If the probabilistic distribution of X1 is known, is it possible to derive the the probabilistic distribution of X3?

• This question is better suited for: stats.stackexchange.com. If you ask it there, you will likely be luckier, because there you will find more people which are capable of answering this question. – nbro Feb 11 '19 at 11:12
• Anyway, I don't really think you can know the probability distribution of $X_3$ without knowing the functions $f$ and $g$, which define how $X_3$ is dependent on $X_1$ and $X_2$, but I'm not an expert on this topic. – nbro Feb 11 '19 at 11:38

## 2 Answers

Yes you can, provided you know about $$f$$ and $$g$$. Expression $$X3 = f(X1, g(X1))$$can be written as $$X3 = h(X1)$$ where $$h$$ takes into account both $$f$$ and $$g$$. After this finding the PDF is simple by differentiating the CDF: $$F_{X3} (x3) = P(X3 \leq x3) = P(h(X1) \leq x3) = P(X1 \leq h^{-1}(x3))$$

$$\frac {d F_{X3} (x3)}{dx3} = \frac {d P(X1 \leq h^{-1}(x3))}{dx3} = f_{X3}(x3)$$

NOTE: The conventions followed are the same as used in the field of Probablity

(Take care of the function inversion step in non-monotonic cases)

Check these lectures.

No, it is not possible. We could derive the most probable $$x_3$$ by calculating the maximum likelihood: $$x^*_3=\underset{x_3}{\arg\max} p(x_1,x_2|x_3)$$. We are unable to calculate this as you only stated that there is a correlation, but we don't know how it looks like.