Suppose we have 2 Erlang random variables, X and Y, that are independent. Let Z = Y - X and assume that the rate parameter for both X and Y is 1. X ~ Erlang(1, 1) and Y ~ Erlang(3,1).
From my intuition, I think that Z ~ Erlang(2, 1). This means that the moment generating function of Z is ($\frac{1}{1-t}$)$^2$
However, when I try to derive this I get stuck. This is what I have
E[$e^{-tZ}$] = E[$e^{-t(Y - X)}$] = E[$e^{-tY}$]*E[$e^{+tX}$] = ($\frac{1}{1-t}$)$^3$ * ($\frac{1}{1+t}$)
This is not equivalent not ($\frac{1}{1-t}$)$^2$ which means that either my intuition is wrong or I did something wrong in the derivation. Could someone kindly point out to me what I did wrong with an explanation?
Your intuition is wrong.
Notice that $Z=Y-X$ is a random variable here with: $$P(Z<0)=P(X>Y)>0$$so cannot have Erlang-distribution.