← Stat 110: Probability

Unit 6: Moments and Moment Generating Functions

A single transform that packages every moment of a distribution at once, and turns adding independent random variables into multiplying functions.

6 topics · 19 questions

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6.1 Defining the Moment Generating Function

  1. How it worksHow does differentiating the MGF at t equals 0 pull out each moment one at a time?
  2. Worked exampleFind the MGF of a Bernoulli random variable with success probability p.
  3. Going furtherWhat happens to a distribution's MGF when the random variable is shifted by adding a constant?
Your turn1 of 2
Write the MGF of a Bernoulli random variable with success probability p, as a function of t.

6.1 The MGF Uniquely Determines a Distribution

  1. How it worksHow does the one-to-one correspondence between MGFs and distributions turn identification into algebra?
  2. Worked exampleFind the distribution of a random variable whose MGF matches (0.7 plus 0.3et) to the 20th power.
  3. Ties togetherHow does MGF uniqueness explain why a sum of independent Normals is itself Normal?
Your turn1 of 2
A random variable's MGF is (1-0.619 + 0.619et)22. By MGF uniqueness, this identifies a Binomial(n, p) distribution. Find its variance.

6.2 Using the MGF to Find Moments

  1. How it worksHow does the Taylor series of etX explain why each derivative of the MGF pulls out one moment?
  2. Worked exampleFind E[X] and E[X2] for a random variable with MGF M(t) equal to e3t plus 2t^2 by differentiating.
  3. Common mistakeWhy doesn't an MGF need to converge for every real number t to still exist as a function?
Your turn1 of 2
A random variable has MGF M(t) = e6t + 3.42t^2. Find E[X2] by differentiating M twice and evaluating at t = 0.

6.2 MGFs of Sums of Independent Random Variables

  1. How it worksHow does independence let E[et(X+Y)] split into E[etX] times E[etY]?
  2. Worked exampleFind the MGF of the sum of two independent Poisson random variables with rates 3 and 5.
  3. Common mistakeWhy doesn't recognizing the MGF of a sum as a known distribution require redoing a convolution integral?
  4. Ties togetherHow does the sum-of-Poissons MGF trick connect back to a sum of independent Poissons being Poisson?
Your turn1 of 2
Two independent Poisson random variables have rates r1 and r2. Write the MGF of their sum, as a function of t.

6.3 MGFs of the Normal and Exponential Distributions

  1. How it worksHow does completing the square inside the Normal's defining integral produce its MGF?
  2. Worked exampleFind the mean and variance of a Normal distribution from its MGF M(t) equal to e2t plus 4.5t^2.
  3. Going furtherWhat happens to the exponential distribution's MGF as the rate parameter lambda approaches t itself?
Your turn1 of 2
Write the MGF of a Normal distribution with mean 2 and variance 7, as a function of t.

6.3 Moments of the Poisson From Its MGF

  1. How it worksHow does differentiating the Poisson MGF elambda(e^t minus 1) twice recover its mean and variance?
  2. Worked exampleFind E[X] and Var(X) for a Poisson distribution with lambda equal to 7 using its MGF.
  3. Going furtherWhat happens to the relative size of the Poisson's standard deviation compared to its mean as lambda grows large?
Your turn1 of 2
Using the Poisson MGF, find E[X(X-1)] for a Poisson distribution with lambda equal to 11.