STATS

Underwriters at a bank have estimated the probability that two personal mortgage loans will default in the next year, arranged in the probability table below - the random variable X1 denotes the status of customer 1's loan, X2 denotes the status of customer 2's loan, and each random variable is 1 if the loan defaults and 0 otherwise.

	X2 = 0	X2 = 1
X1 = 0	0.94	0.04
X1 = 1	0.01	0.01

 

1. What is the probability that loan 1 defaults?

 

2. Based on these probabilities do the underwriters consider the two random variables  X1 and X2 to be independent?

 

3. The same bank also issued an auto loan to customers 1 and 2. Historically, the bank knows that 80% of customers who defaulted on their mortgages missed a car payment, while only 5% of customers who did not default on their mortgages miss a car payment. Customer 1 missed a car payment this month. After getting this new information, what is the probability (approximately) that he will default on his mortgage?

 

4. Now suppose it was customer 2 who missed a car payment this month. What is the probability that he will default on his mortgage in the next year?