Question

5. Suppose X is a normally distributed random variable with mean μ and variance 2. Consider a new random variable, W=2X + 3.

i. What is E(W)?

ii. What is Var(W)?

6. Suppose the random variables X and Y are jointly distributed. Define a new random variable, W=2X+3Y.

i. What is Var(W)?

ii. What is Var(W) if X and Y are independent?

Answer 1

5.

$E(X)=\mu, V(X)=\sigma^2$

W=2X+3

E(W)=E(2X+3)

=E(2X)+E(3)

=2E(X)+3

$=2\mu+3$

V(W)=V(2X+3)

= V(2X)+V(3)

=4V(X)+0

  $=4\sigma^2+0$

$=4\sigma^2$

#variance of constant is 0.

# V(aX)=a^2V(X)

As per the HOMEWORKLIB RULES I have solved the first question.