Your Textbook Defined the Covariance Between X and Y as Follows

question 17

Essay

Your textbook defined the covariance between X and Y as follows: $$\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$$
Prove that this is identical to the following alternative specification:
$$\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \bar { X } \bar { Y }$$

---

Definitions:

Self-Reference Effect

This refers to the tendency for individuals to better remember information that is personally relevant to them, making it more likely to be recalled than non-self-relevant information.

Self-Esteem

An individual's subjective evaluation of their own worth or value.

Life Goals

Long-term aims that guide individuals' actions and decisions, shaping their life paths and personal fulfillment.

Personal Events

Significant occurrences in an individual's life that have personal importance or impact.