Calculating $\mathbb{P}(Y \leq 1)$ given the moment generating function

Question

Given the moment generating function $$M_Y(t) =\frac{4-3t}{2(t-2)(t-1)}$$ with $t<1$ find $\mathbb{P}(Y \leq 1)$.

First I tried to convert this to the probability generating function, because than you can easily find it, but my TA told me that it's not possible since it's continuous.

Now I've been staring at this for a while and played with the definition of the moment generating function for a bit, but I don't see how I can find $\mathbb{P}(Y \leq 1)$.

Some hint in the right direction is much appreciated :)

Answer 1

Hints:

• Try decomposing your $M_Y(t)$ into partial fractions
• The moment generating function of a mixture distribution, where $Y\sim X_1$ with probability $p$ and $Y\sim X_0$ with probability $1-p$, is $M_Y(t) = pM_{X_1}(t)+(1-p)M_{X_0}(t)$
• The moment generating function of an exponential distribution with rate parameter $\lambda$ is $\dfrac{\lambda}{\lambda-t}$ for $t \lt \lambda$