Let $X\sim N(12,4)$ and $Y \sim N(3,1)$
Let $Z = X – Y$
Find the Moment Generating Function of $Z$.
I tried finding the expected value of $e$ to the power of $tz$, but this isn't possible to separate in the expected value function. I know how to use find the MGF when it is a sum of 2 random variables, but what is the technique when it is a difference like this?
Best Answer
Continuing from @whuber's comment, $-Y$ has normal distribution with mean $-3$ and variance $1$. So $Z = X - Y = X + (-Y)$ has normal distribution with mean $12-3=9$ and variance $4+1=5$. The moment generating function of a normal distribution with mean $\mu$ and variance $\sigma^2$ is $e^{\mu t + \sigma^2 t^2/2}$, and so the moment generating function of $Z$ is $$ e^{9t + \frac52 t^2}. $$