Guess the distribution from the Moment Generating function

Question

If the MGF of a random variable X is
M_x(t) = $\frac{1}{3t}(e^t-e^{-2t})$ for $t\neq0$
Find the distribution of X.

I am currently blank as to how to approach this, a hint would be appreciated

Answer 1

Familiarity with common mgf would help.

Suppose $Y\sim [a,b]$,

$$M_Y(t)=E[e^{tY}]=\int_a^b e^{ty} \frac1{b-a}\, dy=\frac{(e^{tb}-e^{ta})}{(b-a)t}$$

Compare with your question and you should be abe to answer the question.