Economics Assignment
1. The joint probability distribution of X and Y is shown in the table below:
|
|
Y
|
|
X
|
1
|
2
|
|
1
|
0.42
|
0.28
|
|
2
|
0.12
|
0.08
|
|
3
|
0.06
|
0.04
|
a. Determine the marginal distributions of X and Y.
b. Find the conditional probability distribution of X given Y=1
c. Find the conditional mean, variance and standard deviation of X conditional on Y=1
d. Are X and Y independent of each other? Why?
e. Compute the covariance and coefficient of correlation between X and Y.
f. Develop the probability distribution of Z=X+Y
g. Find the mean, variance and standard deviation of Z=X+Y
h. Develop the probability distribution of W=X-Y
i. Find the mean, variance and standard deviation of W= X-Y
j. Develop the joint probability distribution of and Z and W
k. Are Z and W independent of each other? Why?
l. Compute the covariance and coefficient of correlation between Z and W
m. Find the conditional mean and variance of Z conditional on W=0.
2. Suppose that X and Y are random variables. Let W=X+Y and Z=X-Y. Suppose that E[W] =1, E[Z] = 1, V[W] = 7, V[Z] = 3, and V[X] = 4 V[Y]. Find E[X], E[Y], V[X], V[Y], Cov(X,Y) and Cov(W,Z).
3. Two sections of probability course are being taught. From what she has heard about the two instructors listed, Francesca estimates that her chance of passing the class are 0.85 if she gets professor X and 0.6 if she gets professor Y. the section into which she is put is determined randomly by the registrar. Suppose that her chances of being assigned to professor X are 0.4. By the end of the semester we learn that Francesca passes the course. What is the probability she was enrolled in professor X's section?
4. There are two cutting fabric machines in a textile plant. The probability that the first machine will break down on a certain day is 0.09, while the probability that the second machine will break down on a certain day is 0.07. The probability that at least one of the machines will break down on a certain day is 0.14. Find the probabilities of the following events:
a) The first machine will work while the second machine will break down on a given day.
b) The two machines will work on a given day.
c) At least one of the machines will work on a given day.