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If I'm given a joint distribution of 2 random variables say A and B, how would I find the covariance of A,B?

Example joint distribution:

   A  1  2

1    .5 .2

2    .2 .1
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$Cov (X,Y)=E((X-\mu)(Y-\nu))=E(XY-\mu Y-\nu X+\mu\nu)=E(XY)-\mu E(Y)-\nu E(X)+\nu \mu=E(XY)-\mu\nu=E(XY)-E(X)E(Y)$

(Using the linearity of Expectation.) Here $\mu=E(X),\nu=E(Y)$. E(X) is the expectation of the random variable X.

$E(X)=\sum_{i=1}^{\infty}x_iP(X=x_i)$(When random variable is discrete )

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