Question

Given that μXμX = 2 and σ2X=σX2=15,

a) Let Y = 6X + 143 and find σ2YσY2 : , and μYμY :

b) Let Z = 6X + 57 and find σ2ZσZ2:, and μZμZ:

Answer 1

(a)

Given:

(1)

(2)

Y = 6X + 143 (3)

By Theorem:

(i)

(4)

(ii)

(5)

(iii)

$E(aX+bY)=aE(X)+bE(Y)$ (6)

(iv)

(7)

From (3), we get:

a = 6

b = 143

Substituting in (5), we get:

(8)

Substituting (2), equation(8) becomes:

Substituting in (4), we get:

(9)

E(Y) 6E(X) 143

Substituting (1), equation (9) becomes:

(10)

(b)

Given:

(11)

(12)

Y = 6X + 57 (13)

By Theorem:

(i)

(14)

(ii)

(15)

(iii)

$E(aX+bY)=aE(X)+bE(Y)$ (16)

(iv)

(17)   

From (13), we get:

a = 6

b = 57

Substituting in (15), we get:

(18)

Substituting (2), equation(8) becomes:

Substituting in (4), we get:

$E(Y)=6E(X)+57$ (19)

Substituting (11), equation (19) becomes:

(20)

Thus, Answers to Questions asked:

(a)

(b)