Math 2.0

Probability theory and stochastic processes Proof

Proof of the formula for the moment-generating function of the normal distribution

Builds on Moment-generating functionNormal distributionDensity of the normal distributionProbability density functionLaw of the unconscious statisticianIntegration by substitutionExponential of a sumImproper integral

Proof

We substitute x=μ+σzx = \mu + \sigma z and complete the square in the exponent.

  1. Let μ∈ℝ\mu \in \R, σ2>0\sigma^2 > 0, and σ=σ2>0\sigma = \sqrt{\sigma^2} > 0. Let X∼𝒩(μ,σ2)X \sim \mathcal{N}(\mu, \sigma^2) be normal, and fix t∈ℝt \in \R.
  2. Let f=φμ,σ2f = \varphi_{\mu, \sigma^2} and φ=φ0,1\varphi = \varphi_{0,1} be the normal densities. By definition, ff is a probability density function of XX.
  3. By the law of the unconscious statistician, if ∫ℝetxf(x) dx<∞\int_\R e^{tx} f(x) \, dx < \infty, then etXe^{tX} is integrable and E[etX]E[e^{tX}] equals this integral.
  4. The integrand is continuous and positive. So the integral is the limit of ∫μ−σRμ+σRetxf(x) dx\int_{\mu - \sigma R}^{\mu + \sigma R} e^{tx} f(x) \, dx as R→∞R \to \infty, an improper integral.
  5. For all real zz, σf(μ+σz)=φ(z)\sigma f(\mu + \sigma z) = \varphi(z). Substitution with x=μ+σzx = \mu + \sigma z gives ∫μ−σRμ+σRetxf(x) dx=∫−RRetμ+tσzφ(z) dz.\int_{\mu - \sigma R}^{\mu + \sigma R} e^{tx} f(x) \, dx = \int_{-R}^{R} e^{t\mu + t\sigma z} \varphi(z) \, dz.
  6. Completing the square gives tσz−z2/2=σ2t2/2−(z−σt)2/2t\sigma z - z^2/2 = \sigma^2 t^2/2 - (z - \sigma t)^2/2.
  7. By step 6 and the exponential of a sum, etμ+tσzφ(z)=eμt+σ2t2/2 φ(z−σt)e^{t\mu + t\sigma z} \varphi(z) = e^{\mu t + \sigma^2 t^2/2} \, \varphi(z - \sigma t).
  8. Substitution with u=z−σtu = z - \sigma t gives ∫−RRφ(z−σt) dz=∫−R−σtR−σtφ(u) du\int_{-R}^{R} \varphi(z - \sigma t) \, dz = \int_{-R - \sigma t}^{R - \sigma t} \varphi(u) \, du.
  9. Since φ\varphi is the density of 𝒩(0,1)\mathcal{N}(0, 1), its integral over ℝ\R is 11. So the integral in step 8 tends to 11 as R→∞R \to \infty.
  10. By steps 4, 5, 7 and 9, ∫ℝetxf(x) dx=eμt+σ2t2/2<∞\int_\R e^{tx} f(x) \, dx = e^{\mu t + \sigma^2 t^2/2} < \infty.
  11. By step 3, the moment-generating function is MX(t)=E[etX]=exp⁡(μt+12σ2t2)M_X(t) = E[e^{tX}] = \exp\left(\mu t + \tfrac{1}{2}\sigma^2 t^2\right).

Idea

Multiplying the standard normal density by esze^{sz} shifts the centre of the Gaussian bell to ss and multiplies it by es2/2e^{s^2/2}. This is completing the square. The shifted bell still has total mass 11.

Reading path

To understand this page from the ground up, read these 287 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof that the exponential series has infinite radius of convergence, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof of the comparison test, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof of integration by substitution, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that sums and products of continuous functions are continuous, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Definition by recursion concept
  4. Element definition
  5. Factorial definition
  6. Group concept
  7. Integers object
  8. Limit point of a set definition
  9. Lipschitz continuity definition
  10. Local maximum definition
  11. Mathematical induction concept
  12. Natural numbers object
  13. Ordered field definition
  14. Partition of an interval definition
  15. Peano axioms concept
  16. Proposition concept
  17. Random experiment concept
  18. Rational numbers object
  19. Real numbers object
  20. Sequence definition
  21. Set bounded above definition
  22. Proof of Pascal's rule from the recursive definition proof
  23. Set definition
  24. Ring definition
  25. Ring of integers statement
  26. Limit of a function definition
  27. Local minimum definition
  28. Archimedean ordered field definition
  29. Successor function definition
  30. Mesh of a partition definition
  31. Refinement of a partition definition
  32. Riemann sum definition
  33. Conjunction definition
  34. Disjunction definition
  35. Implication definition
  36. Negation definition
  37. Predicate definition
  38. The reals form a field statement
  39. Real sequence definition
  40. Subsequence definition
  41. Infimum definition
  42. Supremum definition
  43. Pascal's rule statement
  44. Ordered pair definition
  45. Commutative ring definition
  46. Continuity at a point definition
  47. Derivative definition
  48. Limit at infinity definition
  49. One-sided limit definition
  50. Union definition
  51. Addition of natural numbers definition
  52. Sample space definition
  53. Proof of the Archimedean property in an Archimedean ordered field proof
  54. Riemann integrable function definition
  55. The rationals form a field statement
  56. The successor function is injective statement
  57. Zero is not a successor statement
  58. Biconditional definition
  59. Truth table concept
  60. Modus ponens concept
  61. Existential quantifier definition
  62. Universal quantifier definition
  63. Bounded sequence definition
  64. Cauchy sequence definition
  65. Geometric sequence definition
  66. Limit of a sequence definition
  67. Monotone sequence definition
  68. Series definition
  69. Completeness axiom statement
  70. Cartesian product definition
  71. Proof of the factorial formula for binomial coefficients by induction proof
  72. Continuous function definition
  73. Antiderivative definition
  74. Differentiable function definition
  75. One-sided derivative definition
  76. Archimedean property statement
  77. Proof of the commutativity of addition by induction proof
  78. Subset definition
  79. Order on the natural numbers definition
  80. Riemann integral definition
  81. The rationals form an ordered field statement
  82. Propositional logic concept
  83. Upper and lower Darboux sums definition
  84. Mathematical proof concept
  85. Uniqueness quantifier definition
  86. Proof of the sum of a finite geometric series proof
  87. Convergent sequence definition
  88. Pointwise convergence definition
  89. Sequential continuity definition
  90. Cauchy product definition
  91. Partial sum definition
  92. Power series definition
  93. Factorial formula for binomial coefficients statement
  94. Proof that a composition of continuous functions is continuous proof
  95. Binary relation definition
  96. Derivative function definition
  97. Universal set definition
  98. Order on the reals definition
  99. Improper integral definition
  100. Power set definition
  101. Commutativity of addition statement
  102. Definite integral definition
  103. Proof of strong induction from ordinary induction proof
  104. Upper and lower Darboux integrals definition
  105. Tautology definition
  106. Uniform convergence definition
  107. Sum of a finite geometric series statement
  108. Convergent series definition
  109. Radius of convergence definition
  110. Proof of the binomial theorem by induction proof
  111. Antisymmetric relation definition
  112. Reflexive relation definition
  113. Transitive relation definition
  114. Higher derivative definition
  115. Complement definition
  116. Interval definition
  117. Function definition
  118. Strong induction statement
  119. Uniform convergence on a set definition
  120. Logical equivalence definition
  121. Sum of a series definition
  122. Series of functions definition
  123. Absolute convergence definition
  124. Binomial theorem statement
  125. Preorder definition
  126. Continuously differentiable function definition
  127. Proof of the epsilon-delta form of continuity proof
  128. Proof of the epsilon-delta form of the limit of a function proof
  129. Sigma-algebra definition
  130. Injective function definition
  131. Indicator function definition
  132. Preimage of a set definition
  133. Proof of the well-ordering principle from strong induction proof
  134. Proof of the double negation law by a truth table proof
  135. Partial order definition
  136. Epsilon-delta definition of continuity statement
  137. Epsilon-delta definition of a limit statement
  138. Countable set definition
  139. Event definition
  140. Well-ordering principle statement
  141. Borel sigma-algebra definition
  142. Double negation law statement
  143. Total order definition
  144. Uniform continuity definition
  145. Composition of continuous functions is continuous statement
  146. Probability axioms concept
  147. Proof by contradiction concept
  148. Proof that every sequence has a monotone subsequence, by peaks proof
  149. Uniform continuity on a set definition
  150. Proof that Lipschitz functions are uniformly continuous proof
  151. Probability measure definition
  152. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  153. Every sequence has a monotone subsequence statement
  154. Lipschitz functions are uniformly continuous statement
  155. Probability distribution definition
  156. Dirac measure definition
  157. Probability space definition
  158. The reals form an ordered field statement
  159. Absolute value definition
  160. Random variable definition
  161. Proof of the approximation property of the supremum proof
  162. Cumulative distribution function definition
  163. Law of a random variable definition
  164. Discrete random variable definition
  165. Proof of the triangle inequality for real numbers proof
  166. Proof that the absolute value is multiplicative proof
  167. Random variable with a density definition
  168. Proof of the epsilon-N definition of the limit of a sequence proof
  169. Approximation property of the supremum statement
  170. Probability mass function definition
  171. Triangle inequality statement
  172. Absolute value is multiplicative statement
  173. Probability density function definition
  174. Expected value definition
  175. Proof of the intermediate value theorem by the supremum proof
  176. Proof that the reals are Archimedean from the completeness axiom proof
  177. Epsilon-N definition of the limit of a sequence statement
  178. Proof of the reverse triangle inequality proof
  179. Law of the unconscious statistician statement
  180. Proof that differentiable functions are continuous proof
  181. Density of the normal distribution definition
  182. Intermediate value theorem statement
  183. Proof of the squeeze theorem for sequences proof
  184. Proof that limits preserve non-strict inequalities proof
  185. Proof of the limit of a sum of sequences proof
  186. Proof of the uniqueness of limits proof
  187. Proof of the linearity of the integral proof
  188. The reals are Archimedean statement
  189. Proof that convergent sequences are bounded proof
  190. Proof that convergent sequences are Cauchy proof
  191. Proof that Cauchy sequences are bounded proof
  192. Proof of the monotone convergence theorem for sequences proof
  193. Reverse triangle inequality statement
  194. Moment definition
  195. Normal distribution definition
  196. Differentiable functions are continuous statement
  197. Proof that there is a natural number n with 1/n below any positive epsilon proof
  198. Linearity of the integral statement
  199. Proof of the monotonicity of the integral proof
  200. Limits preserve non-strict inequalities statement
  201. Proof that Darboux and Riemann integrals agree proof
  202. Limit of a sum of sequences statement
  203. Proof of the ratio formula for the radius of convergence proof
  204. Uniqueness of limits statement
  205. Convergent sequences are bounded statement
  206. Squeeze theorem statement
  207. Convergent sequences are Cauchy statement
  208. Cauchy sequences are bounded statement
  209. Monotone convergence theorem for sequences statement
  210. Moment-generating function definition
  211. Proof of the limit of a product of sequences proof
  212. There is a natural number n with 1/n below any positive epsilon statement
  213. Monotonicity of the integral statement
  214. Darboux and Riemann integrals agree statement
  215. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  216. Proof that a decreasing sequence bounded below converges proof
  217. The sequence r^n tends to zero for |r| < 1 example
  218. Proof of the sequential characterization of limits proof
  219. Proof that continuity is equivalent to sequential continuity proof
  220. Decreasing sequence bounded below converges statement
  221. Limit of a product of sequences statement
  222. Proof of Riemann's criterion for integrability proof
  223. Geometric series example
  224. A series of nonnegative terms converges iff its partial sums are bounded statement
  225. Continuity is equivalent to sequential continuity statement
  226. Riemann's criterion for integrability statement
  227. Sequential characterization of limits statement
  228. Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
  229. Proof of the ratio test proof
  230. Proof of the comparison test proof
  231. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  232. Proof of the chain rule with an auxiliary difference quotient proof
  233. Proof of the linearity of differentiation proof
  234. Comparison test statement
  235. Proof that sums and products of continuous functions are continuous proof
  236. Proof of the additivity of the integral over intervals proof
  237. Cauchy product of absolutely convergent series statement
  238. Bolzano–Weierstrass theorem statement
  239. Chain rule statement
  240. Product of continuous functions is continuous statement
  241. Linearity of differentiation statement
  242. Proof that continuous functions on a closed bounded interval are uniformly continuous proof
  243. Ratio test statement
  244. Additivity of the integral over intervals statement
  245. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  246. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  247. Proof of the constant multiple rule proof
  248. Continuous functions on a closed bounded interval are uniformly continuous statement
  249. Boundedness theorem statement
  250. Proof of the first fundamental theorem of calculus proof
  251. Ratio formula for the radius of convergence statement
  252. Cauchy criterion for sequences statement
  253. Constant multiple rule statement
  254. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  255. Uniformly Cauchy sequence of functions definition
  256. Proof of the Cauchy criterion for series proof
  257. Extreme value theorem statement
  258. Proof of Fermat's theorem on stationary points proof
  259. Cauchy criterion for series statement
  260. Proof that continuous functions are Riemann integrable proof
  261. Fermat's theorem on stationary points statement
  262. Proof that absolutely convergent series converge proof
  263. Continuous functions are Riemann integrable statement
  264. Proof of Rolle's theorem proof
  265. Absolutely convergent series converge statement
  266. Rolle's theorem statement
  267. Proof of the mean value theorem for integrals proof
  268. Proof of the Weierstrass M-test proof
  269. Proof of the mean value theorem from Rolle's theorem proof
  270. Mean value theorem for integrals statement
  271. Weierstrass M-test statement
  272. Mean value theorem statement
  273. First fundamental theorem of calculus statement
  274. Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
  275. Proof that the integral of a continuous function is an antiderivative proof
  276. Proof of the second fundamental theorem of calculus proof
  277. Power series converge uniformly on smaller discs inside the radius of convergence statement
  278. The integral of a continuous function is an antiderivative statement
  279. Second fundamental theorem of calculus statement
  280. Proof that the exponential series has infinite radius of convergence proof
  281. Proof of integration by substitution proof
  282. The exponential series has infinite radius of convergence statement
  283. Integration by substitution statement
  284. Exponential function definition
  285. Proof of the exponential of a sum by the Cauchy product proof
  286. Exponential of a sum statement
  287. Proof of the formula for the moment-generating function of the normal distribution proof

Where this leads

20 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_distribution, Moment- and cumulant-generating functions
About this page
TagVKP5HGD7TV
URIm2://VKP5HGD7TV
Kindproof
WritingW0 stub
MSC60E10
ProvesMoment-generating function of the normal distribution
UsesMoment-generating function, Normal distribution, Density of the normal distribution, Probability density function, Law of the unconscious statistician, Integration by substitution, Exponential of a sum, Improper integral
Changed4 November 2026

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