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 0.17 0.20 0.10 10 0.01 0.15 0.11 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 =

from given data:

marginal distribution of x:

 x P(x) xP(x) x^2P(x) 0 0.210 0.000 0.000 5 0.510 2.550 12.750 10 0.280 2.800 28.000 total 1.000 5.350 40.750 E(x) = 5.3500 E(x^2) = 40.7500 Var(x)= E(x^2)-(E(x))^2= 12.1275

marginal distribution of Y:

 y P(y) yP(y) y^2P(y) 0 0.080 0.000 0.000 5 0.380 1.900 9.500 10 0.330 3.300 33.000 15 0.210 3.150 47.250 total 1.000 8.350 89.750 E(y) = 8.3500 E(y^2) = 89.7500 Var(y)=σy= E(y^2)-(E(y))^2= 20.0275

E(XY) =41.75

a)

 Covar(y,x)=E(XY)-E(X)*E(Y)= -2.9225

b)

 Correlation coefficient ρ=Cov(X,Y)/√(σx*σy)= -0.1875

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