So I came across this video on which states Co-Varince of $X$ and $Y$ as: $$cov(X,Y) = \sigma_X \sigma_Y$$ I have not come across this formulae anywhere before. The closest is when we define: $$ \rho = \frac{cov(X,Y)}{\sigma_X \sigma_Y}$$ and if $\rho = 1$ then the former formulae holds true. What exactly am I missing here?
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That formula doesn't hold true. It is just horrible "notation" or a "writeo" or whatever. The formula shown for covariance in terms of summation is correct. Apparently covariance of $X$ and $Y$ is denoted $\sigma_X \sigma_Y$ in the video, but that is just terrible, because it looks like the standard deviations of $X$ and $Y$ are being multiplied, which is not the case. However, $\sigma_{X,Y}$ is often uised to denote the covariance of $X$ and $Y$.
Also, he said the variance is larger than the standard deviation, which is not true when the standard deviation is less than 1.
My advice: Don't use these videos.
