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1) The covariance is \( \mathrm{Cov}(X,Y)=E[XY]-E[X]E[Y] \) (assuming finite moments).
2) The units of covariance are the product of the units of \(X\) and \(Y\).
3) If \( \rho=0 \), then \(X\) and \(Y\) are independent for any distributions.
4) The correlation is \( \rho=\dfrac{\mathrm{Cov}(X,Y)}{\sigma_X\sigma_Y} \) when \( \sigma_X,\sigma_Y>0 \).
5) Correlation is always between \(-1\) and \(+1\) inclusive.
6) Correlation is always strictly between \(-1\) and \(+1\) (i.e., it cannot equal \(\pm1\)).
7) If \( \mathrm{Cov}(X,Y)=0 \), then \(X\) and \(Y\) must be independent.
8) If \(X\) and \(Y\) are independent, then \( \mathrm{Cov}(X,Y)=0 \).
9) Adding a constant to \(X\) changes the correlation between \(X\) and \(Y\).
10) Correlation is unitless.
11) The slope of the least-squares regression of \(Y\) on \(X\) equals the correlation \( \rho \).
12) If \( Y=aX+b \) with \(a>0\), then \( \rho<1 \) whenever \(b\ne0\).
13) For any random variables with finite variances, \( -1 \le \mathrm{Cov}(X,Y) \le 1 \).
14) \( \mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\rho \) (no other factors needed).
15) For constants \(a,b\), \( \mathrm{Cov}(aX,bY)=ab\,\mathrm{Cov}(X,Y) \).