Let ZN01 Create an rv YN14 as a simple looking function of Z

Let Z~N(0,1). Create an r.v. Y~N(1,4), as a simple looking function of Z. Make sure to check that your Y has the correct mean and variance.

We will have to use basic properties of mean and variances .

Properties.

E(aX) = aE(X)

E(X+b) = E(X) +b

Var(aX) = a²Var(X)

Var(X+b) = Var(X)

Using these properties if ,

Y = 2Z +1

The

E(Y) = E(2Z+1) = E(2Z) +1 = 2E(Z) +1 = 0

Var(Y) = Var(2Z+1) = Var(2Z) = 2²Var(Z) = 4*1 = 4.

This our random variable Y must be such that,

Y = 2Z +1