Sample Spaces

1. Suppose S is a uniform sample space with N elements. If E is any possible come and ω is the probability function for S evaluate ω(e).

2. Define a probability function on the set A = {1, 2, 3} such that A is not a uniform sample space.

3. Given the sample space B = {a, b, c} and probability function ω on B. If ω(a) = 0.3, ω({a, b}) = 0.8 then find ω(b) and ω(c).

4. Suppose that only 30% of a birds hatchlings survive their first year. If a bird hatches 7 chicks, what is the probability exactly 3 will survive their first year? What is the probability at most 3 will survive?

1. Consider a stock with the following return information for 2015-2020

Year    Return

2015    25%

2016    -15%

2017    3%

2018    41%

2019    -6%

2020    12%

Find the following for your sample of returns: Average Return, Variance, and Standard Deviation.

2. Jamie’s portfolio consists of $15,000 in Microsoft stock and $45,000 in Coca Cola. Microsoft has an expected return of 19% annually with a standard deviation of 42%. Coca Cola has an expected return of 10% annually with a standard deviation of 20%. The correlation between the 2 stocks has been 0.3.

a) Calculate the expected return of Jamie’s portfolio

b) Calculate the standard deviation of Jamie’s portfolio

You are given the following balance sheet information about Bank of the Coyote. Use it to answer the questions.

a. Calculate Bank of the Coyote's leverage ratio.

b. Suppose the Bank of the Coyote earned $0.8 million in after-tax profits. Calculate the ROA for Bank of the Coyote.

c. Calculate Bank of the Coyote's ROE.

Consider a continuous culture of Bacillus subtilis bacteria, which are used to produce the enzyme amylase during the growth phase. The relevant parameters for growth of this bacteria are μ_MAX = 0.88 hr^-1, Y_X/S = 0.5 gdw/gr substrate, S₀ = 4.0 gr substrate/L, and the Monod constant K_S = 0.3 gr/L, while the protein yield during growth phase is Y_P/X (aka product parameter α) = 0.12 g protein/gdw. The chemostat is operated at a dilution rate that is 60% of the maximum growth rate. Using this information, what is the steady state rate of product formation in the chemostat culture, in units of grams protein/L-hour?

How do I find T score and Stanine when (Xbar = 66.73, σ = 5.77)

2. Adapted from Exercise 3.19 in the textbook. The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days.

(a) Between what range of days do about 95% of pregnancies last?

(b) What proportion of pregnancies last more than 170 days?

(c) What proportion of pregnancies last less than 245 days? 1

(d) What proportion of pregnancies last between 245 and 280 days?

(e) A pregnancy that lasts an amount of time 0.3 standard deviations less than the means lasted for how many days?

Consider the following scenario analysis:

Assume a portfolio with weights of 0.60 in stocks and 0.40 in bonds.

a.What is rate of return on the portfolio in each scenario? (Enter your answer as a percent rounded to 1 decimal place.)

Recession rate of return:?%

Normal Economy rate of return:?%

Boom rate of return:?%

b. What are the expected rate of return and standard deviation of the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Expected rate of return:?%

Standart deviation:?%

Donald and Hillary each have 10 hours of labor (L) that can be used to produce candy (C) and soup (S). Donald’s utility function is given by: Ud = C^0.3 S^0.7 and Hillary’s is: UH = C^0.5S^0.5. Neither have preferences over production. Production of each good is determined by: C = 2Lc and S = 3Ls.

(a) What must be the price ratio pc/ps be?

(b) Set the wage equal to $1. Given the above price ratio, how much C and S will Donald and Hillary demand?

(c) How should labor be allocated between the two goods to meet the demand calculated in part b)?

Star Corp has 5 million shares outstanding and no growth opportunities. It just paid a dividend of $6 per share and has a market value of $62.5 million.

a. What is the discount rate on Star’s stock?

b. The firm has an opportunity to invest in a new project. The project requires an additional investment of $1.2 million today and will generate additional earnings in the next 10 years. The first earnings amount of $0.3 million will begin one year from now and grow at 4% each year thereafter (until year 10). Assuming no taxes, calculate the NPV of the new project.

c. What will the stock price be if the firm undertakes the new project?

You are researching a population of brittle stars that varies in the number of legs that they produce. You survey the local population, and come up with the numbers below. Assume that the phenotypes you encounter represent the full range of possible phenotypes in this population.

1. Calculate the phenotypic variance in this trait.
2. Do the traits in this population appear to be normally distributed? Give an explanation of why this pattern might be observed.

3. Through a series of breeding experiments, you calculate that h2 for this population is 0.3. If no dominant alleles contribute to this trait, and there are no epistatic interactions involving the gene loci that control this trait, what is VE for this population?

#of legs: # of individuals:

5 ; 420

6 ; 189

7 ; 72

8 ; 5

9 ; 3