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

 Find the probability that a randomly selected program will have neither of these types of errors.

 Find the probability that a randomly selected program will contain at least one syntax error and at most one error in logic.

 Find the marginal densities for X and Y.

Find the probability that a randomly selected program contains at least two syntax errors.

 Find the probability that a randomly selected program contains one or two errors in logic.

 Are X and Y independent?

Find E[X], E[Y], E[XY], and Cov(X,Y)

Find Var(X), Var(Y), and ρ_XY

 Find the condition density of Y given that X = 2, and the conditional mean of Y given that X = 2.