An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.
y | |||||
p(x, y) |
0 | 5 | 10 | 15 | |
x | 0 | 0.03 | 0.06 | 0.02 | 0.10 |
5 | 0.04 | 0.15 | 0.20 | 0.10 | |
10 | 0.01 | 0.15 | 0.13 | 0.01 |
(a) Compute the covariance for X and Y. (Round
your answer to two decimal places.)
Cov(X, Y) =
(b) Compute ρ for X and Y. (Round your
answer to two decimal places.)
ρ =
y | |||||
x | 0 | 5 | 10 | 15 | Total |
0 | 0.0300 | 0.0600 | 0.0200 | 0.1000 | 0.2100 |
5 | 0.0400 | 0.1500 | 0.2000 | 0.1000 | 0.4900 |
10 | 0.0100 | 0.1500 | 0.1300 | 0.0100 | 0.3000 |
Total | 0.0800 | 0.3600 | 0.3500 | 0.2100 | 1.0000 |
marginal distribution of X:
x | P(x) | xP(x) | x^2P(x) |
0 | 0.2100 | 0.0000 | 0.0000 |
5 | 0.4900 | 2.4500 | 12.2500 |
10 | 0.3000 | 3.0000 | 30.0000 |
total | 1 | 5.45 | 42.25 |
E(x) | = | 5.4500 | |
E(x^2) | = | 42.2500 | |
Var(x) | E(x^2)-(E(x))^2 | 12.5475 |
marginal distribution of Y:
y | P(y) | yP(y) | y^2P(y) |
0 | 0.0800 | 0.0000 | 0.0000 |
5 | 0.3600 | 1.8000 | 9.0000 |
10 | 0.3500 | 3.5000 | 35.0000 |
15 | 0.2100 | 3.1500 | 47.2500 |
total | 1.0000 | 8.4500 | 91.2500 |
E(y) | = | 8.4500 | |
E(y^2) | = | 91.2500 | |
Var(y) | E(y^2)-(E(y))^2 | 19.8475 |
a) covariance for X and Y =E(XY)-E(X)*E(Y) = -2.80
b)
ρ for X and Y =Cov(X,Y)/sqrt(Var(X)*Var(Y))= -0.18
Get Answers For Free
Most questions answered within 1 hours.