这道题是在问常数a对covariance 和correlation的分别的影响吗？结果就是对correlation没有影响，对covariance放大a倍？

NO.PZ2020010304000034 问题如下 Whis the effeon the sample covarianbetween X anY if X is scalea constant Whis the effeon the sample correlation? The sample covarianis finen∑i=1(Xi−X‾)(Yi−Y‾)\frac{1}{n}\sum_{i=1} (X_i - \overline X)(Y_i - \overline Y)n1​∑i=1​(Xi​−X)(Yi​−Y).Multiplying eaXi a constant a will rescale the sample mea so that1n∑i=1(aXi−aX‾)(Yi−Y‾)=1n∑i=1a(Xi−X‾)(Yi−Y‾)=a1n∑i=1(Xi−X‾)(Yi−Y‾)\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) n1​∑i=1​(aXi​−aX)(Yi​−Y)=n1​∑i=1​a(Xi​−X)(Yi​−Y)=an1​∑i=1​(Xi​−X)(Yi​−Y)The fincovarianis a times the origincovariance. The correlation is then ρ=σXYσXσY\rho = \frac{\sigma_{XY}}{\sigma_X\sigma_Y}ρ=σX​σY​σXY​​ so thscaling XiX_iXi​ a will proρ=aσXYaσXσY=σXYσXσY\rho = \frac{a\sigma_{XY}}{a\sigma_X\sigma_Y} = \frac{\sigma_{XY}}{\sigma_X\sigma_Y}ρ=aσX​σY​aσXY​​=σX​σY​σXY​​ anthe samle correlation is invariant to rescaling the t 解析中的相关系数公式和讲义78页不一致，请问是怎么推导成的？