Question

4. Over the last 10 years, a company's annual earnings increased year over year seven times and decreased year over year three times. You decide to model the number of earnings increases for the next decade as a binomial random variable. A. What is your estimate of the probability of success, defined as an increase in annual earnings? For Parts B, C, and D of this problem, assume the estimated probability is the actual probability for the next decade. B. What is the probability that earnings will increase in exactly 5 of the next 10 years? C. Calculate the expected number of yearly earnings increases during the next 10 years. D. Calculate the variance and standard deviation of the number of yearly earnings in creases during the next 10 years. E. The expression for the probability function of a binomial random variable depends on two major assumptions. In the context of this problem, what must you assume about annual earnings increases to apply the binomial distribution in Part B? What reserva- tions might you have about the validity of these assumptions?

Answer 1

A.

Estimate of probability of success = Number of years the company’s annual earnings increased / Total number of years

= 7/10 = 0.7

B.

Let X be the number of years the company’s annual earnings increased.

X ~ Binomial(n = 10, p = 0.7)

P(X = 5) = 10C5 * 0.7^5 * (1 - 0.7)^5

= 252 * 0.7^5 * 0.3^5

= 0.1029193

C.

Expected number of yearly earnings increases = np = 10 * 0.7 = 7

D.

Variance of yearly earnings increases = np(1-p) = 10 * 0.7 * (1-0.7) = 2.1

Standard deviation of yearly earnings increases = $\sqrt{2.1}$ = 1.449138

E.

The two major assumptions for the annual earnings increases to follow binomial distribution are,

• The probability of success (annual earnings increases) is the same every year.
• The outcome on annual earnings in a year does not affect the outcome on annual earnings in other years.