.The following table displays the joint probability distribution of two discrete random variables X and Y.

1. What is P(X=1/Y=1)?   
2. What is the value of E(X/Y=1)?   
3. What is the value of VAR(X/Y = 1)?
4. What is the correlation between X and Y?
5. What is variance of W = 4X - 2Y.
6. What is covariance between X and W?

A worker has asked her supervisor for a confidential letter of recommendation for a new job. She estimates that there is an 80% chance that she will get the job if she receives a strong recommendation, a 40% chance if she receives a moderately good recommendation, and a 10% if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate and weak are 0.6, 0.3 and 0.1 respectively. Given that she fails to get the job, what is the probability that she received a weak recommendation?

a) What is the probability that a hand of 13 cards contains four of a kind (e.g., four 5’s, four Kings, four aces, etc.)?

b) A single card is randomly drawn from a thoroughly shuffled deck of 52 cards. What is the probability that the drawn card will be either a diamond or a queen?

c) The probability that the events A and B both occur is 0.3. The individual probabilities of the events A and B are 0.7 and 0.5. What is the probability that neither event A nor event B occurs?

Assuming that 1.00 L of 4% PVA solution is spread out to form a thin sheet of thickness equal to the average length of a PVA strand, calculate the total area of the liquid sheet. Then calculate the average of the sheet of solution per strand of PVA. Assuming this area is a little square, the square root of this area gives the edge length of the square which is approximately the average distance between the PVA strands. Show that this distance is about ten times the estimated diameter of a PVA strand (0.2-0.3 nm).

A curve of radius 30 m is banked so that a 950 kg car travelling at
40 km/h can go round it even if the road is so icy that the coefficient of static friction is approximately zero. You are commissioned to tell the local police the range of speeds at which a car can travel around this curve without skidding. Neglect the effects of air drag and rolling friction. If the coefficient of static friction is 0.3, what is the range of speeds you tell them?

For a newsvendor product the probability distribution of demand X (in units) is as follows:

xi 0 1 2 3 4 5 6
pi 0.05 0.1 0.2 0.3 0.2 0.1 0.05

The newsvendor orders Q = 4 units.

a) Derive the probability distributions and the cumulative distribution functions of lost sales as well as leftover inventory.

b) Knowing that the expected total cost function is convex in the order quantity Q, demonstrate that Q = 4 gives the minimal expected total cost.

Stocks A and B have the following returns in each of the states given below.

The probability of the boom is 0.5, the probability of the normal economy is 0.3 and the probability of the recession is 0.2.
(a) Calculate the variance of the returns of A and the variance of the returns of B
(b) What is the covariance between the returns of A and B?

(c) What is the standard deviation of a portfolio of A and B with equal amounts invested in both?

On 1 June2020, Purchase Limited enters into a firm commitment Supply Limited to buy USD 100,000 of
inventory. On 1 July 2020, the Purchase Limited enters into a hedging arrangement which meets the hedge
accounting criteria stipulated by the accounting standards (Australian Accounting Standards Board (AASB) 9).
Purchase Limited has designated the firm commitment hedging arrangement as a fair value hedge. On 1 August
2020, Supply Limited transfers the inventory to Purchase Limited, and on that date, the Purchase Limited makes
the payment. The spot and forward rates are as follows.

Required:
a) Explain at least two determinants of determining an effectiveness of a hedge instrument against a
hedge
5 Marks
b) Provide journal entries to account for the hedged item (firm commitment to buy inventory) 8 Marks
i. On 1 June 2020
ii. On 30 June 2020
iii. On 1 August 2020
c) Provide journal entries to account for the hedge instrument (forward contract) 7 Marks
i. On 1 June 2020
ii. On 30 June 2020
iii. On 1 August 2020

1. Approximately 25% of the adult population is allergic to pets with fur or feathers, but only 4% of the adult population has a food allergy. A quarter of those with food allergies also have pet allergies.What is the probability a person has food allergies but is not allergic to pets?

A. 0.01

B. 0.03

C. 0.04

D. 0.0625

E. 0.24

2. Suppose the probability there are children in a car involved in an auto accident is 0.3. Further suppose that if there are children in a car that is involved in an auto accident, there is a 0.1 probability the driver was 55 years or older. However, if there no children in a car that is involve in an auto accident, suppose there is a 0.25 probability that driver was 55 years or older.What is the probability there were children in a car involved in an auto accident if the driver was not 55 years or older?

A. 0.66

B. 0.34

C. 0.333

D. 0.27

E. 0.01

3. Suppose the probability there are children in a car involved in an auto accident is 0.3. Further suppose that if there are children in a car that is involved in an auto accident, there is a 0.1 probability the driver was 55 years or older. However, if there no children in a car that is involve in an auto accident, suppose there is a 0.25 probability that driver was 55 years or older.What is the probability there were no children in a car involved in an auto accident if the driver was not 55 years or older?

A. 0.933

B. 0.66

C. 0.525

D. 0.34

E. 0.25