Derive E[X^2] from mgf e^{t^2}

Suppose the moment-generating function is $M_X(t) = E[e^{tX}] = e^{t^2}$. Identify the distribution of $X$ and compute $E[X^2]$. Present the derivative-based calculation using $M_X''(0)$, and explain how it agrees with the variance formula.

Overview: This question assesses knowledge of moment-generating functions, computing moments through derivatives, and how raw moments relate to variance, demonstrating skill in probabilistic characterization and moment calculations.