Answers.Body: <p>No, it is not possible. We could derive the most probable <span class="math-container">$x_3$</span> by calculating the maximum likelihood: <span class="math-container">$x^*_3=\underset{x_3}{\arg\max} p(x_1,x_2|x_3)$</span>. We are unable to calculate this as you only stated that there is a correlation, but we don't know how it looks like.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-02-11T22:43:41.570Z
Answers.LastActivityDate: 2019-02-11T22:43:41.570Z
Answers.UserDisplayName: Benedikt S. Vogler
Body: <p>In the diagram below, there are three variables: <code>X3</code> is a function of (depends on) <code>X1</code> and <code>X2</code>, <code>X2</code> also depends on <code>X1</code>. More specifically, <code>X3 = f(X1, x2)</code> and <code>X2 = g(X1)</code>. Therefore, <code>X3 = f(X1, g(X1))</code>.</p>
  <p><a href="https://i.sstatic.net/Av2hx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Av2hx.png" alt="enter image description here" /></a></p>
  <p>If the probabilistic distribution of <code>X1</code> is known, is it possible to derive the probabilistic distribution of <code>X3</code>?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-11T11:38:45.217Z
Comments.Text: I don't 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 $X_2$.
Comments.UserDisplayName: nbro
Tags: <machine-learning><probability-distribution><statistical-ai><bayesian-networks>
Title: Can we derive the distribution of a random variable based on a dependent random variable's distribution?
UserDisplayName: user297850
ThreadText: # Can we derive the distribution of a random variable based on a dependent random variable's distribution?
  
  (Asked by: username_0 on 2019-02-11)
  
  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))`.
  
  [![enter image description here](https://i.sstatic.net/Av2hx.png)](https://i.sstatic.net/Av2hx.png)
  
  If the probabilistic distribution of `X1` is known, is it possible to derive the probabilistic distribution of `X3`?
  
  **Tags:** machine-learning, probability-distribution, statistical-ai, bayesian-networks
  
  > **Comments:**
  > **username_1**: I don't 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 $X_2$.  
  
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  ## 1 answers
  
  ### Answer by username_2 on 2019-02-11. Score: 0
  
  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.
  
  
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