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I have difficulty visualizing what a joint pdf of two independent random variables might look like. The one that I can think of is a cylindrical extension of a univariate Gaussian (let's say extent from x to y). However, in this case, only X is independent of Y (in the sense that the choice of Y does not affect the distribution of X) but Y is entirely dependent on X (in the same sense).

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Independence is symmetric, as you can see both from the definition $P(X\cap Y)=P(X)P(Y)$ and from the symmetric form of Bayes’ theorem, $P(A\mid B)P(B)=P(B\mid A)P(A)$.

An example of independent variables that might help to visualize the concept is afforded by two univariate Gaussians with different variances. The product of their probability density functions has an elliptically shaped bulge; along each line parallel to an axis it yields a scaled version of the marginal density for that axis.

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  • $\begingroup$ But if say the choice of x will scale the distribution of y (given fixed x), it feels like y is dependent on x. $\endgroup$
    – Sam
    Mar 26, 2020 at 9:37
  • $\begingroup$ @Zzy1130: I'm not sure what that means. Could you elaborate? $\endgroup$
    – joriki
    Mar 26, 2020 at 9:38
  • $\begingroup$ For example in that elliptically shaped bulge, if I fix x=10, I will get a y distribution, I fix x=20, I will get another y distribution. If x and y are independent, this two independent should look identical. But I think it's not the case for an elliptically shaped bulge. $\endgroup$
    – Sam
    Mar 27, 2020 at 12:51
  • $\begingroup$ @Zzy1130: Well, it depends on the elliptically shaped bulge. If the eliptically shaped bulge arises, as in my example, from multiplying the two marginal probability density functions, then the two sections will be similar by construction. For example, look at this eliptically shaped bulge. $\endgroup$
    – joriki
    Mar 27, 2020 at 12:58
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    $\begingroup$ Oh yes let me think through again. thanks for pointing that out. $\endgroup$
    – Sam
    Apr 10, 2020 at 9:06

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