I understand concepts more with visualizations. So I made a Venn diagram for statistically independent, uncorrelated and orthogonal random variables. But I am in a confusion which of the below Venn diagram is correct. Venn diagram-A or Venn diagram-B? Or is there any mistake in the illustration. Also considered here is the case of two random variables.

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    $\begingroup$ Hint: $E[XY]=0$ and $E[XY]=E[X]E[Y]$ both will hold if at least one of $X$ and $Y$ has zero mean. Also, note that statistically independent random variables for which the expectation is undefined or does not exist (e.g. Cauchy random variables) cannot be said to be uncorrelated or orthogonal since it is not possible to talk of expectations at all. $\endgroup$ – Dilip Sarwate Jan 3 '14 at 17:12
  • $\begingroup$ How are these graphics supposed to work? By definition, a Venn diagram represents sets and their subsets. Which sets are represented here? Ordered pairs $(X,Y)$ of random variables, each of which has a density function and a well-defined expectation, perhaps? What does "$f(xy)$" mean? How are $(x,y)$ and $(X,Y)$ related? $\endgroup$ – whuber Jan 5 '14 at 16:06
  • $\begingroup$ OK, sorry my ignorance. $\endgroup$ – dexterdev Jan 6 '14 at 5:34

You can visualize independence of random events $A,B \subset \Omega$ by Venn diagrams. From $P(A)P(B) = P(A\cap B)$ follow the conditions $P(A|B)=P(A)$ and $P(B|A)=P(B)$. The first condition means that the area of A relative to the area of $\Omega$ is the same as the area of $A\cap B$ relatively to the area of $B$.

Now, random variables are basically mappings from subsets of a measurable space $\Omega$ to subsets of a space with known structure, e.g. the real line. Independence of random variables means that all preimages of all (measurable) subsets in $\mathbb{R}$ are independent. That's why Venn diagrams are no longer appropriate, except for Bernoulli random variables.

For correlation and orthogonality, you'll have to take all the real values of a random variable into account. That's why Venn diagrams are no longer instructive at all. Correlation and orthogonality are properties of random variables, independence is a property of their preimages.


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