公式

问题如下：

What is the effect on the sample covariance between X and Y if X is scaled by a constant a?

What is the effect on the sample correlation?

选项：

解释：

The sample covariance is defined

$\frac{1}{n}\sum_{i=1} (X_i - \overline X)(Y_i - \overline Y)$.

Multiplying each Xi by a constant a will rescale the sample mean by a so that

$\frac{1}{n}\sum_{i = 1}(aX_i - a\overline X)(Y_i - \overline Y) =\frac{1}{n}\sum_{i = 1} a(X_i - \overline X)(Y_i - \overline Y) =a\frac{1}{n}\sum_{i = 1} (X_i - \overline X)(Y_i - \overline Y)$

The final covariance is a times the original covariance. The correlation is then $\rho = \frac{\sigma_{XY}}{\sigma_X\sigma_Y}$ so that scaling $X_i$ by a will produce $\rho = \frac{a\sigma_{XY}}{a\sigma_X\sigma_Y} = \frac{\sigma_{XY}}{\sigma_X\sigma_Y}$ and the samle correlation is invariant to rescaling the data.