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 .
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 .
Given below is a bivariate distribution for the random variables x and y. f(x, y) x y 0.3...
Given below is a bivariate distribution for the random variables x and y. f(x,y) x y 0.1 90 70 0.5 20 30 0.3 40 60 a. Compute the expected value and the variance for x and y. b. Develop a probability distribution for x + y. c. Using the result of part (b), compute E(x+y) and Var (x+y). d. Compute the covariance and correlation for x and y. Are x and y positively related, negatively related or unrelated? e. Is...
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