14 questions linked to/from Covariance and independence?
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### Covariance matrices and independency [duplicate]

If we have a diagonal covariance matrix does that guarantee independency?
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### Why zero correlation does not necessarily imply independence

If two variables have 0 correlation, why are they not necessarily independent? Are zero correlated variables independent under special circumstances ? If possible, I am looking for an intuitive ...
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### Simple examples of uncorrelated but not independent $X$ and $Y$

Any hard-working student is a counterexample to "all students are lazy". What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"?
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### Can somebody illustrate how there can be dependence and zero covariance?

Can somebody illustrate, as Greg does, but in more detail, how random variables can be dependent, but have zero covariance? Greg, a poster here, gives an example using a circle here. Can somebody ...
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I am trying to find out the difference between treating two random variables as poisson distributions, $\mathrm{Poisson}(x_1)$, $\mathrm{Poisson}(x_2)$, and using a bivariate poisson, $\mathrm{... 2 votes 1 answer 2k views ### Covariance Zero Equals Independent? We know that$cov(X,Y)=0$does not warranty$X$and$Y$are independent. But if they are independent, their covariance must be$0$. My question is: what kind of distribution must$X$and$Y$be for ... 1 vote 0 answers 631 views ### Interpretation of (diagonalized) inverse covariance matrix There are several threads here about covariance matrix and inverse covariance matrix interpretation (here, here or here). However, I was wondering how to interpret the inverse covariance matrix (or ... 1 vote 1 answer 258 views ### What is wrong with this proof/derivation? Regarding simple linear regression$y = a + bx + \epsilon$where$\epsilon$is uncorrelated, E$[\epsilon]=0$, and Var$[\epsilon]=\sigma^2$, the definition of the residual sum of squares is$...
For sochastic processes $X_n$ and $Y_n$ it holds that if they are indenpendant for all $n$ then $$E[X_nY_n] = E[X_n]E[Y_n]$$ But can you go the other way. I mean if $E[X_nY_n] = E[X_n]E[Y_n]$, does ...