# Questions tagged [independence]

For questions involving the notion of independence of events, of independence of collections of events, or of independence of random variables. Use this tag along with (probability) or (probability-theory). Do not use for linear independence of vectors and such.

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### Paradox with deterministic colliders in directed acyclical graphs

I encounter a paradox with a collider in a DAG (directed acyclical graph) and would appreciate some guidance. In this case the collider $Y$ is a deterministic function of variable $X$ and a second set ...
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### “Gram-Schmidt”-like decomposition method for creating independent variables.

Considering $X$ and $Y$ two variables "unlinked", i.e there is no deterministic $f$ function so that $Y = f(X)$, is there any theorem ensuring existence/characterization of some $g, h$ ...
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### If concentration inequalities correspond to non-asymptotic probability, what do anti-concentration inequalities correspond to?

Concentration of measures is a niche in statistical probability theory, most notable for deriving concentration inequalities. I'm still trying to grasp its underlying motivation and premise compared ...
27 views

### Proof of the product rule in probability theory for causal independence

There is a common attitude in the text books on probability that the so-called product rule is an obvious property, when events are independent, i.e., $P(A\cap B)=P(A)P(B)$ when $A$ and $B$ are ...
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### A normal random variable independent of each component of a multivariate normal random vector.

Suppose that $\mathbf{Y}\sim N_3(0,\,\sigma^2\mathbf{I}_3)$ and that $Y_0$ is $N(0,\,\sigma_0^2)$, independently of the $Y_i$'s. My question is that does $(\mathbf{Y}, Y_0)$ also have multivariate ...
I have a few examples I could provide, here is one. Let X = Uniform(-1,1) and X = $Y^2$. Here we have clearly E(X) = 0 and then E(Y) = $\frac{1}{3}$. Cov(X,Y) = E(X*Y) - E(X) * E(Y) = E($X^3$) - 0 = E(...