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I'm not sure but for proofing marginal independence I've came across this stack exchange post, and from there to this link, where I found that the condition to hold for 2 variables to be independent marginally is $P(X \lor Y) = P(X)$ which I assume can be basically translated to the probability of either events X, or Y happening is identical to probability of X happening which doesn't make much sense. Like that would imply that $P(x) + P(y) - P(x)\times{P(y)} = P(x)$, which is only possible when $P(x)\times{P(y)} = P(y)$ which then implies that $P(x) = 1$ which implies X is certain. So for y to be independent of X, X should be certain? that looks like I've completely messed my math somehow

Nikolai Savulkin
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  • You seem to mix $P(X\ or\ Y)$ with $P_Y(X)$. We do not have $P(X\ or Y)=P(X)$ in the case of independent $X$ and $Y$, think for example of two coin tosses : The probability that at least one of the tosses shows "heads" is $\frac{3}{4}$ and not $\frac{1}{2}$ – Peter Oct 03 '21 at 12:26
  • @Peter thanks, indeed my silly mistake. Thank you for pointing my attention to it. Could you make it an answer so i can accept it? – Nikolai Savulkin Oct 03 '21 at 12:30

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