The Textbook Formula for the Variance of the Discrete Random $$\sigma _ { Y } ^ { 2 } = \sum _ { i = 1 } ^ { k } \left( y _ { i } - \mu _ { Y } \right) ^ { 2 } p _ { i } .$$

question 17

Essay

The textbook formula for the variance of the discrete random variable Y is given as $$\sigma _ { Y } ^ { 2 } = \sum _ { i = 1 } ^ { k } \left( y _ { i } - \mu _ { Y } \right) ^ { 2 } p _ { i } .$$
Another commonly used formulation is
$$\sigma _ { Y } ^ { 2 } = \sum _ { i = 1 } ^ { k } y _ { i } ^ { 2 } p _ { i } - \mu _ { y } ^ { 2 } .$$
Prove that the two formulas are the same.

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