## The king has only one sibling. What is the probability that the sibling is male?

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1 (Probability review) The king has only one sibling. What is the probability that the sibling is male? Assume that every birth results in a boy with probability 1/2, independent of other births. Be careful to state any additional assumptions you have to make in order to arrive at an answer.

2 (Probability review) Sport Obermeyer need to decide how many parkas to produce for Winter 2019. Demand for their parkas follows a Normal distribution with mean 1200 and standard deviation 130.

(a) Suppose Sport Obermeyer produces 1350 parkas. What is the probability that demand is fully met?

3 (Probability review) Consider constructing a portfolio using three assets: bound A, stock B, and stock C. Their corresponding expected returns over the investment horizon are µA = 5%, µB = 15%, µC = 25% The corresponding standard deviations of the returns are σA = 3%, σB = 10%, σC = 30%. The correlation between these assets are ρAB = −0.1, ρAC = −0.5, ρBC = 0.

(a) Consider a portfolio where the investor put 90% of the money in stock B and 10% of the money in Stock C, i.e., P = 0.9B + 0.1C. What is the portfolio’s expected return and standard deviation? (b) Suppose the investor has a risk tolerance of σP = 25%, i.e., the portfolio’s standard deviation can not exceed 25%. Construct a portfolio that achieves the highest expected return under this risk constraint. (You may need to use R or other computing software to help you solve this)

(b) How should Sport Obermeyer choose production level x in order to meet demand with probability 0.95?

(c) Sport Obermeyer earns \$14.50 in marginal profit for each unit sold and loses \$ 5.00 for each unit produced but not sold. What is the optimal production level?