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I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa?

Also, if I know that a joint distribution of two variables is a recognized distribution, say the distribution of $X,Y$ is hyper geometric, does that mean that $X\sim HG$ and $Y\sim HG$? what about the other way?

shinzou
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  • You can have dependent but uncorrelated random variables, see e.g. this. The only thing you can say for sure is that independent r.v.'s are uncorrelated. – Clement C. Jun 07 '15 at 17:49
  • Independent implies uncorrelated, so correlated implies not independent (I use dependent for something more causal) – Henry Jun 07 '15 at 17:51

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Let $X$ be $-1$, $0$, or $1$, each with probability $1/3$, and let $Y=X^2$. Then $X$ and $Y$ are uncorrelated, but they are no independed.

Independent random variables for which a correlation exists are always uncorrelated. Their correlation exists only if both of their variances are finite.

I don't know what, if anything, it would mean to say that the joint distribution of two random variables is hypergeometric. The hypergeometric distribution that I know if is a univariate distribution, so $X$ or $Y$ can be hypergeometrically distributed, but the pair $(X,Y)$ cannot.

Michael Hardy
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  • There isn't $(x,y)\sim HG$? – shinzou Jun 07 '15 at 18:26
  • @kuhaku : I don't know what the meaning of that notation would be. I wouldn't be surprised if someone has definied such a thing as a multivariate hypergeometric distribution, but I don't know what the definition is. However, I would be surprised if it means only that each component, $X$ or $Y$, is hypergeometrically distributed. For example, when you say that the pair $(X,Y)$ has a bivariate normal distribution, it means more than just that $X$ and $Y$ each have a univariate normal distribution. ${}\qquad{}$ – Michael Hardy Jun 07 '15 at 19:22
  • What do you mean by it means more? – shinzou Jun 07 '15 at 20:25
  • @kuhaku : I mean that there are some bivariate distributions of pairs $(X,Y)$ that are not bivariate normal distributions but for which $X$ and $Y$ are both normally distributed. For example, suppose $X\sim N(0,1)$ and $\Pr(Y=X)=1/2=\Pr(Y=-X)$. For that it follows that $Y\sim N(0,1)$, but the pair $(X,Y)$ is not very close to a bivariate normal distribution. ${}\qquad{}$ – Michael Hardy Jun 08 '15 at 01:33