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

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.

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

Company: Capital One

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Technical Screen

Given the moment-generating function M_X(t) = E[e^{tX}] = e^{t^2}, identify the distribution of X and compute E[X^2]. Show the derivative-based calculation from M_X′′(0) and reconcile it with the variance formula.

Quick Answer: 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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Oct 13, 2025, 9:49 PM
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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:

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