Question

Given below is a bivariate distribution for the random variables x and y.

f(x, y)	x	y
0.3	50	80
0.2	30	50
0.5	40	60

(a)

Compute the expected value and the variance for x and y.

E(x)

=

E(y)

=

Var(x)

=

Var(y)

=

(b)

Develop a probability distribution for

x + y.

x + y	f(x + y)
130
80
100

(c)

Using the result of part (b), compute

E(x + y)

and

Var(x + y).

E(x + y)

=

Var(x + y)

=

(d)

Compute the covariance and correlation for x and y. (Round your answer for correlation to two decimal places.)

covariancecorrelation

Are x and y positively related, negatively related, or unrelated?

The random variables x and y are  ---Select--- positively related negatively related unrelated .

(e)

Is the variance of the sum of x and y bigger, smaller, or the same as the sum of the individual variances? Why?

The variance of the sum of x and y is  ---Select--- greater than  less than unrelated the sum of the variances by two times the covariance, which occurs whenever two random variables are  ---Select--- positively related negatively related unrelated .

Answer 1

a)

f(x, y)	x	y	xp	x^2p	yp	y^2p
0.3	50	80	15	750	24	1920
0.2	30	50	6	180	10	500
0.5	40	60	20	800	30	1800
		sum	41	1730	64	4220

E(X) = 41

E(Y) = 64

Var(X) = E(X^2) - (E(X))^2

=1730 -41^2

= 49

Var(Y) = 4220-64^2

=124

b)

x+y	p
130	0.3
80	0.2
100	0.5

c)

x+y	p	p(x+y)	p(x+y)^2
130	0.3	39	5070
80	0.2	16	1280
100	0.5	50	5000
		105	11350

E(X+Y) = 105

Var(X+Y) = 11350-105^2

= 325

d)

f(x, y)	x	y	xp	xyp
0.3	50	80		1200
0.2	30	50		300
0.5	40	60		1200
		sum		2700

Cov(X,Y) = E(XY) - E(X)E(Y)

= 2700 - 41 *64

= 76

covariance > 0

hence x and y are positively related

e)

Var(X+Y) = 325

Var(X) +Var(Y) = 49+124

= 173

Var(X+Y) > Var(X) +Var(Y)

The variance of the sum of x and y is greater than unrelated the sum of the variances by two times the covariance, which occurs whenever two random variables are positively related .