Find the Moment Generating Function $Y$. What is the distribution of $Y$?

Question

Let $X_1$ and $X_2$ be independent normal variables with means 2 and 5 and variances 9 and 1.

Let $Y = 3X_1 + 6X_2 - 8$. Find MGF. What is the distribution of $Y$.

attempt:

Im not sure about the MG.

But for the distribution I got N~[28, 109]. Does this answer this question?

Answer 1

Sure the variance is $117$ as it is said in André´s comment. I think you have rather miscalculated it than not knowing how to calculate it. It seems that you substracted the constant $8$. But the constant does not affect the variance.

The moment generating function of the normal distribution can be looked up. Let denote the moment generating function of $X$ as $M_X(t)$, where $X$ is normal distributed. Similar for the random variable $Y$. And let $Z=a+bX+cY$

Then the MFG of $Z$ is

$e^{at}\cdot M_X(tb)\cdot M_Y(tc)$

Answer 2

Comments:

Looking through the Q and A here, I'm not absolutely sure what prompted the question. Not truly realizing that this kind of linear combination of normal random variables is again normal? Not realizing that once the mean and variance of a normal RV are known it is trivial to write the MGF? In any case, the Answer from @callculus should cover most of the technical details (+1).

In case it might help with intuition or visualization, here is a simulation in R of a million realizations of $Z$.

[Notes: In R the parameters of a normal distribution are the mean $\mu$ and the SD $\sigma.$ The null hypothesis of Shapiro-Wilk test is that the data are from some normal distribution. The implementation of this test in R will handle up to 5000 observations.]

 m = 10^6;  x1 = rnorm(m, 2, 3);  x2  = rnorm(m, 5, 1)
 z = 3*x1 + 6*x2 - 8
 mean(z);  sd(z)
 ## 27.98318  # aprx pop mean (about 2-3 signif digits)
 ## 10.81479  # aprx pop SD (about 2-3 signif digits)
 shapiro.test(z[1:5000])

    Shapiro-Wilk normality test

 data:  z[1:5000]      # normality test of 1st 5000 simulated z's
 W = 0.9998, p-value = 0.878  # cannot reject normality

The following histogram shows a good match of the million simulated $Z$'s with the normal distribution having $\mu = 28$ andd $\sigma = \sqrt{117} \approx 10.817,$