Hello I have a question in regard to moment generating functions and something I noticed but wasnt to clear on.
Say that a random variable $X$ has $$MGF=M_{X}(t)=\frac{4}{4-t^2}$$ for $-2 \lt t \lt 2$
and we were asked, what is the moment generating function of $cX$ or in particular $1.2X$
But when I had originally seen the question asked I tried to use that $$E(X)=M'_{X}(0)=0$$ and thus (using wolfram for exact results) $$Var(X)=E(X^2)=M''_{X}(0)=\frac{1}{2}$$ And noting that $Var(1.2X)=(1.2)^2Var(X)=\frac{18}{25}=0.72000...$
Now, the answer that was given as the correct one was, $$M_{1.2X}(t)=\frac{4}{4-1.44t^2}$$
For clarity say $Y=1.2X$ However, if I use the moment generating function on that I get for $E(Y)=0$ and so $Var(Y)=E(Y^2)$which using wolfram results in a non rational number of $0.719998$
So where is my mistake in rounding? Is this just a matter of rounding error or precision? Or is there some big mistake I have or misunderstanding?
Moreover, I am interested in learnig more about the properties of MGF. I know a random variable is uniquelly determiend by its MGF, but I am wondering can just knowing the $E(x)$ and $E(x^2)$ be enough or whats important is that it is of all orders? Thanks for any help