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	0.17	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 =

Answer 1

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