Is there an alternate way of writing $E[Y]E[X^2] - E[X]E[XY]$?

Question

I've been studying bivariate least squares regression, and I derived the following formula for what the regression line should be: \begin{align} Y = aX + b, \end{align} where \begin{align} a = \frac{E[XY] - E[X]E[Y]}{E[X^2] - E[X]^2}, \end{align} and \begin{align} b = \frac{E[Y]E[X^2] - E[X]E[XY]}{E[X^2] - E[X]^2}. \end{align} The expression for $a$ can be beautifully rewritten as \begin{align} a = \frac{\text{Cov}(X,Y)}{\text{Var}[X]}, \end{align} but I can't see a way to write the numerator of $b$ in a simpler form. Is there a way?

Answer 1

You could write : $$ E[Y]E[X^2] - E[X]E[XY] = (E[Y]E[X^2] - E[Y]E[X]^2) + (E[Y]E[X]^2 - E[X]E[XY]) \\ = E[Y] \mbox{Var}(X) - E[X] \mbox{Cov(X,Y)} $$

for the numerator of $b$. The denominator is $\mbox{Var}(X)$ so you get : $$ b = E[Y] - aE[X] = E[Y-aX] $$

once you take the quotient. The point of writing $b$ in this form, is (roughly) noting that if $Y=aX+b$(were completely true) then $Y-aX = b$, and taking expectation of both sides gives you $E[Y - aX] = b$. Since the error is two-sided, the estimator will be unbiased, so this relation is to be expected.