|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

5. Calculate the conditional probability distribution, ?(?|?)P(A|B).

6. Calculate the conditional probability distribution, ?(?|?)P(B|A).

7. Does ?(?|?)=?(?|?)P(A|B)=P(B|A)? What do we call the belief that these are always equal?

8. Does ?(?)=?(?|?)P(A)=P(A|B)? What does that mean about the independence of ? and B?

What is a probability model? What is the relevance?

The probability that "X is at most 3" is interpreted as:

what is the reason behind the calculation of probability?

Probability          Stock A                 Stock B                 Stock C

.30                          9%                          35%                        3%

.30                          15%                        28%                        9%

.40                          7%                          21%                        15%

6) Find the expected return of the portfolio with 50% invested in stock A and 50% invested in stock B. (Round your answer to 2 decimal places)

7) Find the expected return of the portfolio with 40% invested in stock A, 20% invested in stock B, and 40% invested in stock C. (Round your answer to 2 decimal Places)

8) Find the portfolio 1 Variance (answer 1) and standard Deviation (answer 2). (round to 2 decimal places)

9) Find the portfolio variance (answer 1) and standard deviation (answer 2) for portfolio 2. (round your answer to 2 decimal places.

Why is the Max probability equal to 1

1. A scientist is studying human memory. Subjects are shown a five-digit number for different lengths of time and then must write the number down. The subject either correctly recalls the number or fails to recall it. (Answer using R CODING) The results are as follows:

times <- c(10.73, 9.9, 9.61, 8.7, 8.56, 8.31, 8.18, 7.86, 7.63, 6.99, 6.66, 6.1, 5.92, 5.84, 5.67, 5.64, 5.56, 5.29, 5.1, 5.09, 4.92, 4.81, 2.86, 2.13, 2.05, 1.95, 1.67, 1.67, 1.38, 1.02)

correct <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0)

Here, times shows the amount of time given to the student, and correct is a Bernoulli variable, with 1 meaning the number was correctly recalled 0 meaning it was not.

a) Report your p-value, state your null hypothesis, state whether your p-value is significant, and interpret.

b) give a rough estimate for the time required for the number to be correctly recalled with 90% probability

The state in which you live operates a lottery. The proceeds of the lottery are used to supplement the state’s unemployment insurance fund. You can play the lottery for $1 per chance. To play, you choose a three-digit number from 000 to 999, inclusive, and receive an official ticket with that number printed on it. Each evening, a ball is drawn blindly from a container that holds 1,000 balls, each marked with a different three-digit number. If the number on your ticket is selected in the daily drawing on the date you play, you receive $500 for your ticket. Otherwise, you receive nothing. a. Suppose you buy one ticket. What is the probability that your number will win? b. What is the expected utility (in dollars) of a single play? c. Suppose you decide to play three times in one day, and you choose the same numbers each time. (You hold three tickets at a cost of $3.) What is the expected utility (in dollars) of your triple play? d. Suppose you decide to play three times in one day, and you choose different numbers each time. What is the expected utility (in dollars) of this triple play?

1. Which of the following is not the advantage of VaR?

It is easy to understand.

It shows the expected loss given that the loss is greater than the absolute VaR level.

It is the loss level that will not be exceeded with a certain probability.

It captures an important aspect of risk in a single number.

2.Which of the following bonds has the longest duration?

7-year, 7% coupon bond

7-year, 12% coupon bond

14-year, 7% coupon bond

14-year, 12% coupon bond

3.One investment project has a probability of 0.03 of a loss of $20 million and a probability of 0.97 of a loss of $2 million during a one-year period. What are the 95% VaR and expected shortfall (ES) for this project?

VaR = $2 million; ES = $20 million

VaR = $20 million; ES = $12.8 million

VaR = $2 million; ES = $12.8 million

VaR = $20 million; ES = $0.64 million

4.Suppose that you have a bond position worth $100 million. Your position has a modified duration of 8 years and a convexity of 150. By how much does the value of the position change if interest rates increase by 25 basis points?  Use the duration-convexity rule.

($1,953,125)

($1,906,250)

($2,046,875)

($2,187,500)