Using MGF Technique to prove a theorem

Question

Use the MGF technique to prove the theorem:

If X₁, X₂,...,X_n are independent random variables where X_i~Normal(µ_i, σ_i²), then Y=Σ a_ix_i follows a normal distribution with parameters µ = Σa_iµ_i and σ²= Σa_iσ_i²

I do know how to use MGF for univariate case, but I am having a hard time understanding how to translate this to multivariate

Answer 1

Statement $µ_y = \sum_i a_iµ_i$ is correct. However, statement $σ_y^2= \sum_i a_iσ_i^2$ is wrong and should be $σ_y^2= \sum_i a_i^2σ_i^2$ because $Var(aX) = a^2Var(X).$

You do not show what you have tried to do with MGFs, but I'm guessing the incorrect statement about the variance of $Y$ may be causing you trouble. If you are still having difficulty with MGFs, please edit your question showing some of your work and where you're stuck, so that one of us can help with that. [To find the MGF of the sum of two independent random variables, take the product of the two MGFs. See my Comment.]

In the example below, independent random variables $X_i$ have $E(X_1) = 50, Var(X_1) = 5^2 = 25;$ $E(X_2) = 60, Var(X_2) = 10^2 = 100;$ $E(X_3) = 60, Var(X_3) = 3^2 = 9.$

Also $Y = 2X_1 + 3X_2 + 4X_3.$ So $E(Y) = 520, Var(Y) = 1144.$ With a million iterations of $Y,$ the sample mean $\bar Y \approx E(Y)$ and the sample variance $S_Y^2 \approx Var(Y).$

set.seed(2021)
x1 = rnorm(10^6, 50, 5)   # mean=50, var=25
x2 = rnorm(10^6, 60, 10)  # mean+60, var=100
x3 = rnorm(10^6, 60, 3)   # mean=60, var=9
y = 2*x1 + 3*x2 + 4*x3
mean(y)
[1] 520.0287   # aprx 2(50)+3(60)+4(60) = 520
var(y)
[1]  1144.222  # aprx 4(25)+9(100)+16(9) = 1144

Because a linear combination of independent normal random variables is normal with means and variances according to the formulas above, a histogram [blue] of the million values of $Y$ is well approximated by the density function [orange] of $\mathsf{Norm}(\mu_Y=520, \sigma_Y=\sqrt{1144}).$

hdr = "Simulated values of Y with Normal Density"
hist(y, prob=T, br=50, col="skyblue2", main=hdr)
curve(dnorm(x, mean(y), sd(y)), add=T, col="orange", lwd=2)