Question

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 =

Answer 1

	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