So I'm revising moment generating functions and I'm stuck on a part of a question I'm looking at.
So I am asked to find the moment generating function of a random variable X whose distribution is given by $$\Bbb{P}(X=1)=\Bbb{P}(X=-1)=1/2$$
I get the Moment generating function for this random variable X to be $$M(t)=\frac{e^t-e^{-t}}{2}$$I am then asked to show that $$M(t)\le \exp\left(\frac{t^2}{2}\right)$$
I am not sure how to go about doing this. I was thinking I could use Taylor series, and expand both sides of the inequality, but I seem to get stuck and can't actually show that one is less than or equal to the other.
Any help appreciated-thanks.
Best Answer
Your moment generating function is wrong. One way you can quickly tell: If $M$ is a moment-generating function, then $M(0)=1.$
It should be:
$$M(t)=\frac{e^t+e^{-t}}{2}=\sum_{k=0}^{\infty} \frac{t^{2k}}{(2k)!}$$
This is because $E\left(X^{2k}\right)=1$ and $E\left(X^{2k+1}\right)=0$ for integers $k\geq 0$.
The we have: $$\exp\left(\frac{t^2}{2}\right)=\sum_{k=0}^{\infty} \frac{t^{2k}}{2^k\cdot k!}$$
So, since $t^2\geq 0,$ if you can prove that $\frac{1}{(2k)!}\leq \frac{1}{2^k\cdot k!}$ for all $k,$ you'd be done. I'll leave that to you.