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.

---

Definitions:

Inelastic Demands

Describes a situation where the demand for a good or service is relatively unresponsive to changes in price, with a percentage change in quantity demanded that is less than the percentage change in price.

Elastic Demands

This refers to the sensitivity of the quantity demanded of a good to a change in its price, where a small price change leads to a larger change in quantity demanded.

Elastic Demands

A situation where the demand for a good or service significantly changes in response to changes in its price.

Close Substitutes

Products or services that can easily replace each other in the eyes of the consumer, offering similar benefits and functionalities.