7 – M4 L2A 07 Taking Constants Out Of Variance And Covariance Optional V3

If you haven’t looked at variance operators in a while, it may not be obvious why we’re able to take the constant factor exposure out by squaring it. Just a reminder of why we’d square the constant when putting it outside of the variance operator. Remember, that variance takes the square of each observations difference from the mean. For example, let’s pretend there are just two data points for factor one with equal probabilities, then the variance of factor one looks like this. If we take the variance of a constant times the variable, then it looks like this. Notice that this constant shows up next to every variable. If we take the constant out one step at a time, then we’ll eventually factor out that constant from the squared terms, which means we’ll also apply the square operator on this constant, then we’ll factor out the squared constant a bit more. Notice that the variance of a constant times a random variable is the same as the squared constant times the variance of that variable. You can pause the video here if you want to look a bit longer. So, that’s why we can take the constant out of variance operator by squaring it. For similar reasons, we can also take the constants out of the covariance operator.