Linear Combination of Independent Random Variables

Question

I am working on a problem and am a bit stuck

The problem: For independent random variables

X ~ N(-1,3)

Y ~ N(0,2)

Z ~ N(4,1)

Consider U:= 2X - 4Y - Z + 5

Find the Expectation, variance, and MGF for U.

What I have done so far:

Fx(t) = Ee$^{tX}$ = exp(-t + $3t^2\over2$)

Fy(t) = Ee$^{tY}$ = exp(0t + t$^2$)

Fz(t) = Ee$^{tZ}$ = exp(4t + $t^2\over2$)

Then MGF of U:

Fu(t) = Ee$^{t(2X-4Y-Z+5)}$ = E[e$^{(2t)X}$e$^{(-4t)Y}$e$^{(-t)Z}$e$^{(5t)}$] =

e$^{(5t)}$Ee$^{(2t)X}$Ee$^{(-4t)Y}$Ee$^{(-t)Z}$ = e$^{(5t)}$Fx(2t)Fy(-4t)Fz(-1t) =

exp(5t) exp(-(2t) + $3(2t)^2\over2$) exp(0(-4t) + (-4t)$^2$) exp(4(-1t) + $(-1t)^2\over2$) =

exp(5t -2t +0t -4t + 6t$^2$ +16t$^2$ + $1\over2$$t^2$) = exp(-1t + 22.5t$^2$) =

U ~ N(-1, 22.5)

Then for Expectation:

F'u(t) =

But here is where I am stuck. How do we find the Expectation of U (how do we take this derivative) and how do we find the Variance of U? Also, does the first part look correct?

Answer 1

Expecation is linear.

$$E[U]= 2E[X] - 4E[Y] - E[Z] + 5$$

Also, since they are independent.

$$Var[U]= 2^2Var[X] + 4^2Var[Y] + Var[Z] $$

Remark:

• Usually we use $F$ for CDF and $M$ for mgf.
• $U$ is again a normal distribution and after you find out the mean and variance, figuring out the mgf should be easy.