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.

Distinguish between entrepreneurs and other innovators.

Understand the concept of optimal R&D expenditures for firms.

Comprehend the relationship between the interest-rate cost of funds and the expected rate of return.

Identify factors that can shift the expected-rate-of-return curve and their impacts on R&D investment.

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Definitions:

Market Price

The current price at which a good or service can be bought or sold in the open market.

Costs Of Production

The total expenses incurred in producing goods or services, including raw materials, labor, and overheads.

Willingness To Sell

The minimum price at which a person or entity is prepared to sell a good or service, reflecting the value at which they are ready to part with it.

Producer Surplus

Producer surplus is the difference between what producers are willing to accept for a good or service versus what they actually receive, indicating the benefit to producers.