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

Last updated: Mar 29, 2026

Quick Overview

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.

Capital One

Oct 13, 2025, 9:49 PM

Data Scientist

Technical Screen

Statistics & Math

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.

Solution

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