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What is a moment generating function and why care?

hard

Answer

  • MX(t)  =  E[e(tX)]M_{X}(t)\; = \;E[e(\mathrm{tX})].
  • When it exists in an open interval around 0, it uniquely determines the distribution — two RVs with the same MGF are identically distributed.
  • Useful because: (1) generates moments via derivatives (MX(0)  =  E[X],  MX(0)  =  E[X2],  )(MX(0)\; = \;E[X], \;MX(0)\; = \;E[X^{2}], \;); (2) MGF of sum of independents = product of MGFs → easy proofs (sum of normals normal, sum of gammas gamma).
  • Characteristic functions (imaginary argument) exist even when MGF doesn't (Cauchy).
Check yourself — multiple choice
  • M(t)  =  E[Xt]M(t)\; = \;E[X^{t}]
  • MX(t)  =  E[e(tX)]M_{X}(t)\; = \;E[e(\mathrm{tX})]; uniquely determines distribution; generates moments; product law under sum of independents
  • Only for uniform
  • Never useful

MGF: E[e(tX)]E[e(\mathrm{tX})]; generates moments; multiplicative under independence.

#probability#moments#theory

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