http://www.econstreams.com/bivariateproof.jpg
Image uploaded to the link above. I'm just not seeing the connection between the 2nd equation on the left handside and the equation on the right. Expanding the terms yields a positive sign for $n\bar X \bar Y$ on the numerator.
Also, the $X_i\bar Y, \bar X Y_i$ terms are uncorrelated, and hence equal 0.
Anyone know how to solve this admittedly very simple proof as sequentially as possible? Any and all help is greatly appreciated! Thanks.
The version on the right is using a shortcut formula for the covariance and variance of a security's returns. The covariance of X and Y is $E(XY)-E(X)E(Y)$, which is the numerator on the left and the right (albeit different forms). Don't forget that the 1/n will cancel out in the fraction, so you are actually seeing sums, not averages, but that is just due to simplifiation.
in the formujla on the rignt, they just multiplied the shortcut formulas by n/n to get the forms you see so as to avoid fractions within fractions.