Question

4. Consider the random variable Z from problem 1, and the random variable X from problem 2.

Also let f(X,Z)represent the joint probability distribution of X and Z.  f is defined as follows:

f(1,-2) = 1/6
f(2,-2) = 2/15
f(3,-2) = 0
f(4,-2) = 0
f(5,-2) = 0
f(6,-2) = 0
f(1,3) = 0
f(2,3) = 1/30
f(3,3) = 1/6
f(4,3) = 0
f(5,3) = 0
f(6,3) = 0
f(1,5) = 0
f(2,5) = 0
f(3,5) = 0
f(4,5) = 1/6
f(5,5) = 1/6
f(6,5) = 1/6

Compute the covariance of X and Z.

Then, compute the correlation coefficient of X and Z. (Note: You will need values that you computed in problems 1 and 2.)

These are questions 1 and 2.

1. Let Z be a random variable with the following probability distribution f:

f(-2) = 0.3
f(3) = 0.2
f(5) = 0.5

Compute the E(Z), Var(Z) and the standard deviation of Z.

2. Tossing a fair die is an experiment that can result in any integer number from 1 to 6 with equal probabilities. Let X be the number of dots on the top face of a die. Compute E(X) and Var(X).

Answer 1

covariance of X and Z = E(XZ) - E(X) E(Z)

f(x,z)	x	z	xzf(x,z)
f(1,-2) = 1/6	1	-2	-2/6
f(2,-2) = 2/15	2	-2	-8/15
f(3,-2) = 0	3	-2	0
f(4,-2) = 0	4	-2	0
f(5,-2) = 0	5	-2	0
f(6,-2) = 0	6	-2	0
f(1,3) = 0	1	3	0
f(2,3) = 1/30	2	3	6/30
f(3,3) = 1/6	3	3	9/6
f(4,3) = 0	4	3	0
f(5,3) = 0	5	3	0
f(6,3) = 0	6	3	0
f(1,5) = 0	1	5	0
f(2,5) = 0	2	5	0
f(3,5) = 0	3	5	0
f(4,5) = 1/6	4	5	20/6
f(5,5) = 1/6	5	5	25/6
f(6,5) = 1/6	6	5	30/6

E(X),E(Z), Var(X), Standard deviation of X and Z are calculated below

E(Z) = 2.5

E(X) = 3.5

standard deviation of Z = 3.0414

standard deviation of X = 1.7078

covariance of X and Z = E(XZ) - E(X) E(Z) = 13.3333 - 2.5 x 3.5 = 13.3333 - 8.75 = 4.5833

covariance of X and Z = 4.5833

Correlation coefficient of  X and Z= 0.8824

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Given,

Z : Random variable with probability distribution

z	f(z)	zf(z)	z2f(z)
-2	0.3	-0.6	1.2
3	0.2	0.6	1.8
5	0.5	2.5	12.5
Total	1	2.5	15.5

Var(Z) = E(Z²) - E(Z)² = 15.5 - 2.5² = 15.5 - 6.25 = 9.25

Var(Z) = 9.25

standard deviation of Z =

X; number of dots on the top face

Probability distribution of X:

X	f(X=x)	xf(x)	x2f(x)
1	1/6	1/6	1/6
2	1/6	2/6	4/6
3	1/6	3/6	9/6
4	1/6	4/6	16/6
5	1/6	5/6	25/6
6	1/6	6/6	36/6
	Total	21/6	91/6

Var(X) = E(X²) - E(X)² = 15.1667 - 3.5² = 15.1667 - 12.25 = 2.9167

Var(X) = 2.9167

Standard deviation of X =