A random variable X has moment generating function $e^{500t+500t^{2}}$ .
Find P(27100 < $(X − 500)^2$ < 50200).
I'm a little confused how exactly I can apply a moment generating function, to which I don't know the distribution, to a central limit-type problem where I want to find the probability that it lies between two quantities. How do I start this problem?
The normal distribution with mean $\mu$ and variance $\sigma^2$ has mgf $$\exp\left(\mu t+\frac{1}{2}\sigma^2 t^2\right).$$ So from your given mgf you can see that $X$ has normal distribution, and can identify the mean and variance of $X$. The rest of the calculation is probably familiar.