My friend and I are going over today's lecture and we can't seem to answer this question: Why can't the second derivative of a moment generating function be negative?
I know it can't be negative at $0$ (this derivative gives $E(X^2)$) but why is it $\ge 0$ in general?
If you believe we can differentiate under the integral then:
$\frac{d^2}{dt^2} \int^{\infty}_{-\infty} e^{tx} f(x) dx = \int^{\infty}_{-\infty} \frac{d^2}{dt^2}( e^{tx} f(x)) dx = \int^{\infty}_{-\infty} x^2 e^{tx} f(x) dx$
Since $x^2e^{tx}f(x) \geq 0$ for all $x$, the integral must be non-negative as well.