Question: A certain company’s board of directors has 13 members. How many ways are there to choose a president, vice president, secretary,

Exercise 1. A certain company’s board of directors has 13 members. How many ways are there to choose a president, vice president, secretary, and treasurer of the board, where no member of the board can hold more than one office?

Exercise 2. The G20 (or Group of Twenty) is an international economic forum for the governments of 19 countries and the European Union. The European Union is itself a block of 28 nations that are located primarily in Europe. How many ways are there to select 11 countries in the G20 to serve on an economic council if 6 are selected from among the 19 member nations not belonging to the European Union and the others are selected from the remaining 28 member nations (those belonging to the European Union)?

Exercise 3. How many different anagrams (including nonsensical words) can be made from the letters in the word STATISTICS, using all the letters?

Exercise 4. Prior knowledge about a stock indicates that the probability θ that the price will rise on any given day is either 0.25 or 0.75. Past data from similar stocks suggest that θ is equally likely to be 0.25 or

0.75 (i.e., P (θ = 0.25) = P (θ = 0.75) = 0.5). Let A be the event that the price of the stock rises on each of 5 consecutive days. Assuming that the price changes are independent across days (so that the probability that the price rises on each of 5 consecutive days is θ⁵), find the probability P (θ = 0.25|A) that θ = 0.25 given 5 consecutive price increases.

Exercise 5. A chartered financial analyst can choose any one of Routes A, B, or C to get to work. The probabilities that he/she will arrive at work on time using Routes A, B, and C are 0.41, 0.57, and 0.64, respectively. Suppose that the probability that he/she chooses Route C is 0.3. Assuming that the analyst is twice as likely to choose Route A as he/she is to choose Route B, and supposing that he/she has just arrived at work on time, what is the probability that he/she chose Route B?

Exercise 6. A modified random walk model for a stock price assumes that at each time step the price can either increase by a fixed amount ∆ > 0, decrease by this same fixed amount ∆ > 0, or remain unchanged. Suppose that P₁, 0 < P₁ < 1 is the probability of an increase, that P₂, 0 < P₂ < 1 is the probability of a decrease, and that P₃ = 1 − P₁ − P₂ is the probability of no change. Let X be the discrete random variable representing the change in a single step.

1. Find the expected value of the random variable X.
2. Find the variance of the random variable X.

Exercise 7. The number of defaults in one year within a certain portfolio of bonds is found to be a Poisson random variable with parameter λ = 5 (i.e., the portfolio has an expectation of 5 defaults per year).

1. Find the probability that the bond portfolio will have fewer than 2 defaults during the upcoming year.
2. Describe the distribution of the continuous random variable representing the time between successive defaults.
3. Find the expected time between successive defaults.
4. Calculate the probability of there being less than 8 months between two successive defaults.

Exercise 8. A biased coin has a 0.6 chance of coming up heads when flipped. Find the probability of flipping 3 or fewer heads in 10 flips. What is the expected number of heads in 10 flips?

Exercise 9. Let X be a random variable representing a quantitative daily market dynamic (such as new information about the economy). Suppose that today’s stock price S₀ for a certain company is $150 and that tomorrow’s price S₁ can be modeled by the equation S₁ = S₀ · e^X . Assume that X is normally distributed with a mean of 0 and a variance of 0.5.

1. Find the probability that X is less than or equal to 0.1.
2. 1.  

   1            2           3            4           5

   Suppose the daily dynamics X_i, i = 1, 2, 3, 4, 5 of each of the next five consecutive days are indepen-
    

   dently and identically distributed as X so that Y

   logarithmic return.

    

   =∆ X + X + X + X + X represents the stock’s five-day

    
   1. 1. Describe the distribution of the continuous random variable Y .
      2. Calculate the probability that Y is less than or equal to 0.1.
    
   2.  

   S

   Describe the distribution of the random variable ^S1

   0

    

   representing the ratio of tomorrow’s price to

    

   today’s price.

3.  

4. S

   What is the expected value of the random variable ^S1 ?

   0

5.  

6. S

   What is the variance of the random variable ^S1 ?

   0

   Exercise 10. Let the dynamics X_i, i = 1, 2, . . . , d be independently and identically distributed as Z ∼

   N (0, 1).  One approach for modeling the short-term interest rate r_t at any time t is given by defining

   r =∆ X² + X² + . . . + X².

   t             1            2                         d

7. Describe the distribution of the continuous random variable r_t.
8. Find the probability that r_t ∈ (0, 0.07] if d = 3.
9. Find the probability that r_t ∈ (0, 0.07] if d = 5.

10. Exercise 11. (Essay Question) Write one page summarizing what you learned in this class. Explain how the fields of Probability and Statistics can be applied to the world of Finance, and describe how you might use these fields in your own chosen career path.