Thread: Proving the equation

Suppose we define

$\displaystyle S_{xy}=\sum\limits_{i=1}^N (y_i-\mu_y)(x_i-\mu_x)/(N-1)$

How do I show that $\displaystyle E(\overline{y}-\mu_y)(\overline{x}-\mu_x)=(1-f)S_{xy}/n$ ?

where $\displaystyle f=n/N$

Originally Posted by noob mathematician

Suppose we define

$\displaystyle S_{xy}=\sum\limits_{i=1}^N (y_i-\mu_y)(x_i-\mu_x)/(N-1)$

How do I show that $\displaystyle E(\overline{y}-\mu_y)(\overline{x}-\mu_x)=(1-f)S_{xy}/n$ ?

where $\displaystyle f=n/N$

you can't

$\displaystyle E(\overline{y}-\mu_y)(\overline{x}-\mu_x)$ is a constant

while $\displaystyle S_{xy}$ is a rv

My guess is that you're mixing up population means (mu's) and sample means (x bar's)

or is that Expectation a sum?