Find probabilities given moment generating function.

Question

The moment generating function of $X$ is $$m(t)= \exp( -6t+32t^2)$$ Find

A. $\;\;P(-4<X<16)$

B. $\;\;P(-10<X<0)$

I have known that the first derivative for moment generating function is the expected value of $X.$

Answer 1

HINT: Observe that the mgf of $X\sim\mathcal{N}(\mu,\sigma^2)$ is $M_X(t)=\textrm{e}^{\mu t+\frac{1}{2}\sigma^2t^2}$. So your random variable is $X\sim\mathcal{N}(-6,8^2)$.