Covariance between data stats problem

[danago]

Hi. At the moment in class we are going over statistics :yuck:

Anyway, the formula I've been using for covariance between two sets of data is:

$$s_{xy} = \frac{1}{n}\sum\limits_{i = 1}^n {x_i y_i } - \overline x \overline y$$


Now, if i was to get a question such as:

"If all the elements of set 'x' are multiplied by 'a', what is the new covariance"

Would this be a valid in mathematical terms:

$$\begin{array}{c} s_{xy} = \frac{1}{n}(ax_1 y_1 + ax_2 y_2 + ... + ax_n y_n ) - a\overline x \overline y \\ = a[\frac{1}{n}(x_1 y_1 + x_2 y_2 + ... + x_n y_n ) - \overline x \overline y ] \\ = a[\frac{1}{n}\sum\limits_{i = 1}^n {x_i y_i } - \overline x \overline y ] \\ \end{array}$$

Therefore, if the elements of the set of data are multiplied by a constant, the covariance is also changed by that same factor? Has my working been valid in terms of the summation notation. The reason i ask is because we've never worked much with summations, so I am not 100% sure how to deal with them.

Thanks,
Dan.

 

[Dick]

Your reasoning is fine. Summations are just sums.

 

[danago]

Alright thanks for the confirmation. I just thought that maybe i was overlooking something, and had possibly made a mathematical error.

Thanks again.
Dan.