7.3 Moment Generating Functions

Moment generating functions (MGFs) are powerful tools in probability theory, providing a compact way to describe random variables. They offer insights into distribution properties, simplify calculations of moments, and help prove important theorems in statistics.

MGFs connect to the broader study of expectation and variance by encoding all moments of a distribution. They allow easy computation of expected values and variances, making them invaluable for analyzing random variables and their transformations in various statistical applications.

Definition and Properties of MGFs

Defining Moment Generating Functions

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• Moment generating function (MGF) represents expected value of $e^{tX}$ where X is a random variable
• Denoted as $M_X(t) = E[e^{tX}]$ for continuous random variables
• For discrete random variables, calculated as $M_X(t) = \sum_{x} e^{tx} p(x)$
• Exists only when expectation is finite for some interval of t around zero
• Provides alternative method to characterize probability distributions

Key Properties and Theorems

• Taylor series expansion of MGF yields moments of the distribution
• MGF expansion: $M_X(t) = 1 + E[X]t + \frac{E[X^2]}{2!}t^2 + \frac{E[X^3]}{3!}t^3 + ...$
• Uniqueness theorem states two distributions with identical MGFs must be the same distribution
• MGF of sum of independent random variables equals product of their individual MGFs
• Scaling property: If Y = aX + b, then $M_Y(t) = e^{bt}M_X(at)$

Relationships to Statistical Measures

• First derivative of MGF at t=0 gives expected value: $E[X] = M'_X(0)$
• Second derivative of MGF at t=0 relates to variance: $Var(X) = M''_X(0) - (M'_X(0))^2$
• Higher-order derivatives at t=0 provide higher moments of the distribution
• MGF simplifies calculations for expected values of functions of random variables

Related Generating Functions

Probability Generating Function

• Probability generating function (PGF) applies to discrete random variables
• Defined as $G_X(s) = E[s^X] = \sum_{k=0}^{\infty} s^k P(X=k)$
• Relates to MGF through $M_X(t) = G_X(e^t)$
• Used to find probabilities and moments of discrete distributions
• Particularly useful for analyzing sums of independent discrete random variables

Characteristic Function

• Characteristic function defined as $\phi_X(t) = E[e^{itX}]$ where i is the imaginary unit
• Always exists for any random variable, unlike MGF
• Fourier transform of the probability density function
• Useful for proving limit theorems (Central Limit Theorem)
• Inversion formula allows recovery of distribution from characteristic function

Joint Moment Generating Function

• Extends MGF concept to multivariate distributions
• For random vector X = (X₁, ..., Xₙ), defined as $M_X(t) = E[e^{t'X}]$ where t' is transpose of t
• Allows computation of mixed moments and joint distributions
• Useful for analyzing dependence between random variables
• Marginal MGFs obtained by setting other variables' t values to zero

Applications and Common Distributions

Probability Theory Applications

• MGFs facilitate proofs of important theorems (Law of Large Numbers)
• Used in deriving distributions of transformed random variables
• Simplify calculations in hypothesis testing and confidence intervals
• Aid in parameter estimation techniques (Method of Moments)
• Provide tools for analyzing stochastic processes (Markov chains)

Moments and Cumulants

• Moments obtained by differentiating MGF and evaluating at t=0
• First moment (mean): $\mu = E[X] = M'_X(0)$
• Second central moment (variance): $\sigma^2 = E[(X-\mu)^2] = M''_X(0) - (M'_X(0))^2$
• Cumulants derived from logarithm of MGF: $K_X(t) = \ln(M_X(t))$
• Cumulants often simplify calculations in probability theory and statistics

MGFs for Common Distributions

• Normal distribution: $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2}$
• Exponential distribution (rate λ): $M_X(t) = \frac{\lambda}{\lambda - t}$ for t < λ
• Poisson distribution (rate λ): $M_X(t) = e^{\lambda(e^t - 1)}$
• Binomial distribution (n trials, probability p): $M_X(t) = (pe^t + 1 - p)^n$
• Gamma distribution (shape k, scale θ): $M_X(t) = (1 - \theta t)^{-k}$ for t < 1/θ