Given below is a bivariate distribution for the random varia

Given below is a bivariate distribution for the random variables x and y. Compute the expected value and the variance for x and y. Develop a probability distribution for x + y. Using the result of part (b), compute E(x + y) and Var (x + y). Compute the covariance and correlation for x and y. Are x and y positively related, negatively related, or unrelated? Is the variance of the sum of x and y bigger, smaller, or the same as the sum of the individual variances? Why?

Solution

a. expected value x = 37; variance of x = 61

expected value of y = 59 ;variance of y = 129;

b)

E(x+Y) =E(X)+E(y) =37+59 = 96

cov(x,y) = 87

Var(X+Y) = Var(X)+Var(Y)+2Cov(X,y) = 61+129+ (2*87) = 364

correlation for x and y = 0.9807;

x and y are positively related

the variance of the sum of x and y is bigger than the individual variances of x, y. since x+y is bigger than individual x and y and the probabilities are the same.

X\\Y	80	50	60
50	0.2
30		0.5
40			0.3