Question 1
Suppose X ~ N(30, 144), and W~N(40,225).
a. If X and W are uncorrelated, find the mean and variance of X + 2W.
b. Find the probability that X + 2W > 120.
Henceforth, suppose that X and W have a correlation coefficient ρ=-.25.
c. What is the covariance of X and W?
d. Find the probability that X + 2W >120.
e. Find the probability that < X + 2W <120
Question 2
Our bank from Question 2 has decided to look more deeply into the matter of customer wait times. In addition to information on the waiting times, the bank has compiled information about the credit scores of the applicants. That is, the bank has 20 observation of the following 2 variables:
|
Observation
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
Wait Time
|
5
|
7
|
22
|
4
|
12
|
9
|
9
|
14
|
3
|
6
|
|
Credit Score
|
740
|
730
|
550
|
700
|
650
|
660
|
630
|
600
|
760
|
730
|
|
Observation
|
11
|
12
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
|
Wait Time
|
5
|
8
|
10
|
17
|
12
|
10
|
9
|
4
|
3
|
13
|
|
Credit Score
|
700
|
620
|
600
|
580
|
650
|
670
|
670
|
790
|
750
|
610
|
a. Find the sample mean and variance of the Credit Score variable (you can call this variable Y if you like).
b. Find the sample covariance and sample correlation coefficient of Wait Times and Credit Scores.
c. Give a short interpretation of the correlation coefficient for this example.
d. What story can you tell that would explain the correlation coefficient the bank observes.