The two most common types of errors made by programmers are syntax errors and errors in logic. for a simple language such as basic the number of such errors is usually small. let x denote the number of syntax errors and y the number of errors in logic made on the first run of a basic program. the sample space for x is {0, 1, 2, 3} and the sample space for y is {0, 1, 2, 3, 4, 5}. assume that the joint density for (x, y) is shown in the following table:

x

y

0

1

2

3

0

.400

.100

.020

.005

1

.300

.040

.010

.004

2

.040

.010

.009

.003

3

.009

.008

.007

.003

4

.008

.007

.005

.002

5

.005

.002

.002

.001

(5 points) find the probability that a randomly selected program will have neither of these types of errors.

(5 points) find the probability that a randomly selected program will contain at least one syntax error and at most one error in logic.

(5 points) find the marginal densities for x and y.

(5 points) find the probability that a randomly selected program contains at least two syntax errors.

(5 points) find the probability that a randomly selected program contains one or two errors in logic.

(5 points) are x and y independent?

(10 points) find e[x], e[y], e[xy], and cov(x, y)

(10 points) find var(x), var(y), and rhoxy

(10 points) find the condition density of y given that x = 2, and the conditional mean of y given that x = 2.