Question

# An instructor has given a short quiz consisting of two parts. For a randomly selected student,...

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

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