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:
$$\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \bar { X } \bar { Y }$$

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The Answer of 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:
$$\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \bar { X } \bar { Y }$$

Your Textbook Defined the Covariance Between X and Y as Follows