Question

Let random variable X be distributed according to the p.m.f P( 0.3 0.5 0.2 · If Y=2X, what are li E r Var(Y) . If Z = aX + b has EZj-0 and Var(Z)-1, what are: lal

Answer 1

Y = 2^X

For X = 1, 2, 3 Y = 2, 4, 8

Thus, the PMF of Y is,

P(Y = 2 ) = 0.3

P(Y = 4 ) = 0.5

P(Y = 8 ) = 0.2

E[Y] = 0.3 * 2 + 0.5 * 4 + 0.2 * 8 = 4.2

E[Y²] = 0.3 * 2² + 0.5 * 4² + 0.2 * 8² = 22

Var[Y] = E[Y²] - (E[Y])² = 22 - 4.2² = 4.36

Z = aX + b

E[X] = 0.3 * 1 + 0.5 * 2 + 0.2 * 3 = 1.9

E[X] = 0.3 * 1² + 0.5 * 2² + 0.2 * 3² = 4.1

Var[Y] = E[Y²] - (E[Y])² = 4.1 - 1.9² = 0.49

E[Z] = 0

=> E[aX + b] = 0

=> aE[X] + b = 0

=> 1.9a + b = 0 ---(1)

Var[Z] = 1

=> Var[aX + b] = 1

=> a²Var[X] + Var[b] = 1

=> a²Var[X] + 0 = 1

=> 0.49a² = 1

=> a² = 1/0.49

=> |a| = 1/0.7 = 10 / 7 = 1.428571

From (1),

1.9a + b = 0

=> b = -1.9 a

=> |b| = 1.9 |a|

=> |b| = 1.9 * 1.428571 = 2.714285