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

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

This question evaluates understanding of moment-generating functions, moment computation via derivatives, and the relationship between raw moments and variance, reflecting competency in probabilistic characterization and moment calculations.

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Question

Identify Distribution and Compute E[X^2] from an MGF

You are given a random variable X with moment-generating function (mgf):

M_X(t) = E[e^{tX}] = e^{t^2}, for all real t.

Tasks:

Identify the distribution of X by comparing M_X(t) to a known family of mgfs.
Compute E[X^2] using the derivative method, i.e., show that E[X^2] = M_X''(0), and display the derivatives.
Reconcile the result with the variance formula E[X^2] = Var(X) + (E[X])^2.

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

Identify Distribution and Compute E[X^2] from an MGF

Solution

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

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Identify Distribution and Compute E[X^2] from an MGF

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