Statistics Assignment_M2

Consider a random experiment in which we roll two fair, FOUR-sided dices, with numbers 1, 2, 3, or 4 on each side. Note that in this experiment there will always one side facing down and the other three sides facing up. Here we will use the number that faces down as the result of rolling a die. Let random variable X be the sum of the two resulting numbers from a roll of the two dices. Let random variable Y be the absolute value of the difference of the two resulting numbers from a roll. For example, suppose if we get “4” on one die and “2” on the other. Then the value of X is 6 and the value of Y is 2. Answer the following questions. (40 pts)

What is the sample space of this random experiment of rolling two fair, four-sided dices? (2 pts)

What is the sample space of X and Y, respectively? (2 pts)

What is the PMF (probability mass function) of X? Let’s denote it as fX(.). (2 pts)

What is the PMF (probability mass function) of Y? Let’s denote it as fY(.). (2 pts)

What is the CDF (cumulative distribution/density function) of X? Let’s denote it as FX(.). (2 pts)

What is the CDF (cumulative distribution/density function) of Y? Let’s denote it as FY(.). (2 pts)

Let A be the event that X is even. What is the value of P(A)? Let B be the event that X≥6. Compute P(B). (2 pts)

Compute P(A∩B) and P(A∪B). (2 points)

Let C be the event that 3≤X≤6. Compute P(C), P(A∩C) and P(A∪C). (3 points)

Are events A and B independent? Why or why not? (2 points)

Are events A and C independent? Why or why not? (2 points)

Compute the expected value (a.k.a. expectation) of X and Y, denoted as E(X) and E(Y), respectively. (2 points)

Compute the variance of X and Y, denoted as Var(X) and Var(Y) or σ_X^2 and σ_Y^2, respectively. One approach is the definition of variance of a random variable. Another approach is making use of the fact that σ_X^2=E(X^2 )-〖[E(X)]〗^2. (2 points)

Develop the joint probability distribution of X and Y using a table. Let f(.) = f(X = a, Y = b) denote the joint probability of X and Y. (4 points)

Find the probability that X is less than or equal to 4 and Y is even. (1 point)

Compute the covariance of X and Y, denoted as cov(X, Y) or σ_XY^ . One approach is the definition. Another approach is making use of the fact that cov(X,Y)=E(XY)-E(X)E(Y). (2 points)

Compute the correlation coefficient of X and Y. Are X and Y correlated (meaning whether X and Y have a linear relationship)? Why or why not? (2 points)

Are X and Y independent? Why or why not? (2 points)

What are the general conclusions about being uncorrelated and independent between any two random variables X and Y? (2 points)

You will use the tabular approach (joint probability table) first and then Bayes’ Theorem to answer problem 42 on page 207 of the textbook. To that end, please follow the steps provided below.

A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 5% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.

Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default.

The bank would like to recall its card if the probability that a customer will default is greater than .20. Should the bank recall its card if the customer misses a monthly payment? Why or why not? (13 points)

Joint Probability Table Approach

Step 1: Define relevant events and collect information

D = default; ND = not default;

M = missing a monthly payment; NM = not miss a payment

Find the probabilities below (3 points):

P(D) = ? P(ND) = ?

P(M|ND) = ? P(NM|ND) = ?

P(M|D) = ? P(NM|D) = ?

Step 2: Complete the Joint Probability Table below. (4 points)

M NM Marginal prob.

Marginal prob. 1

Step 3: Use the information from the joint probability table above in Step 2 to compute the probability that a customer will default if s/he has missed one or more monthly payments. (2 points)

Step 4: Should the bank recall its card if the customer misses a monthly payment? Why or why not? (1 point)

Bayes’ Rule Approach

Use Bayes’ Rule to compute the probability that a customer will default if s/he has missed one or more monthly payments. Show your work. You cannot copy the values of P(D, M) and P(M) from the joint probability table. Instead, use the probability values from Step 1 directly. (3 points)

P(D|M) = P(D, M)/P(M) =

Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, Google’s Chrome browser exceeded a 20% market share for the first time, with a 20.37% share of the browser market. For a randomly selected group of 20 Internet browser users, answer the following questions. (6 points; 2 point each)

Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.