Your Textbook Defines the Correlation Coefficient as Follows Another Textbook Gives an Alternative Formula

question 25

Essay

Your textbook defines the correlation coefficient as follows: $$r = \frac { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } \left( X _ { i } - \bar { X } \right) ^ { 2 } } { \sqrt { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } } \sqrt { \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } }$$ Another textbook gives an alternative formula: $$r = \frac { n \sum _ { i = 1 } ^ { n } Y _ { i } X _ { i } - \left( \sum _ { i = 1 } ^ { n } Y _ { i } \right) \left( \sum _ { i = 1 } ^ { n } X _ { i } \right) } { \sqrt { n \sum _ { i = 1 } ^ { n } Y _ { i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } Y _ { i } \right) ^ { 2 } } \sqrt { n \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } X _ { i } \right) ^ { 2 } } }$$ Prove that the two are the same.
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