Suppose that a real valued random variable with a probability density of $$f(x) = \begin{cases} \frac{1}{8}(x+1) &:-1 <x<5\\ 0 &: \text{else} \end{cases}$$.
I need to verify that $M_{X}(t) \ge e^{t\mu}$, where $M_{X}(t)$ is the moment generating function of X, $t=\frac{1}{2}$ and $\mu$ is the mean of the probability density function, using Jensen's inequality.
From what I understand, I need to show that $E[w(X)] \ge w(\mu)$.
My question is what would I use for the value of $w(X)$? would I use $M_{X}(t)$ or should I use the probability density function? Thanks for any help.
For each $t$, the function $x\mapsto e^{tx}$ is convex. Hence, Jensen's inequality shows that $E[e^{tX}]\geq e^{tE[X]}=e^{t\mu}.$
The problem is pretty straightforward in that the left is $$ \int_{-1}^5 e^{tx}f_X(x)\,dx $$ and the right is just $e^{t\mu}$ where $$ \mu=E[X]=\int_{-1}^5 xf_X(x)\,dx. $$