Question

Use the probability distribution given in the table below and consider two new random​ variables, W​ =  6+7X and V​ =3+1Y​,

to answer the following questions

Joint Distribution of Weather Conditions and Commuting Times
	Rain​(X= 0)	No Rain(X= 1)	Total
Long commute(Y= 0)	0.79	0.03	0.82
Short commute(Y= 1)	0.04	0.14	0.18
Total	0.83	0.17	1.00

A.Compute the mean of W.

E(W)​= (Round your response to two decimal places​)

B.Compute the mean of V.

E(V)​= (Round your response to two decimal places​)

C. Compute the variance of W.

W​= (Round your response to four decimal places​)

D. Compute the variance of V.

V​= (Round your response to four decimal places​)

E.Compute the covariance between W and V.

F.Compute the correlation between W and V.

Answer 1

a) E(W) = E(6 + 7X)

= E(6) + 7E(X)

= 6 + 7 x [(0 x 0.83) + (1 x 0.17)]

= 6 + 7x(0 + 0.17)

= 6 + 1.19

= 7.19 is the answer.

b) E(V) = E(3 + Y)

= E(3) + E(Y)

= 3 + [(0 x 0.82) + (1 x 0.18)]

= 3 + (0 + 0.18)

= 3 + 0.18

= 3.18 is the answer.

c) var(X) = X²P(X) - [XP(X)]²

= E(X²) - [E(X)]²

= [(0 x 0.83) + (1 x 0.17)] - (0.17)²

= 0.17 + 0.0289

= 0.1411

var(W) = var(6 + 7X)

= (7)² x var(X)

= 49 x 0.1411

= 6.9139 is the answer.

d) var(Y) = var(3 + Y)

= var(Y)

= Y²P(Y) - [ YP(Y)]²

= [(0 x 0.82) + (1 x 0.18)] - (0.18)²

= E(Y²) - [E(Y)]²

= 0.18 - 0.0324

= 0.1476 is the answer.