13.1 Moment generating functions for discrete and continuous distributions
3 min read•july 30, 2024
Moment generating functions are powerful tools in probability theory, helping us analyze random variables and their distributions. They uniquely characterize distributions, generate moments, and simplify calculations for sums of independent variables. MGFs are especially useful for proving key theorems and studying linear combinations.
For both discrete and continuous distributions, MGFs are derived using specific formulas involving probability mass or density functions. They allow us to easily calculate moments, variance, skewness, and kurtosis. This approach streamlines complex probability calculations and provides insights into distribution properties.
Moment Generating Functions
Definition and Properties
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() for random variable X defined as [M_X(t) = E[e^{tX}]](https://www.fiveableKeyTerm:m_x(t)_=_e[e^{tx}])
MGFs uniquely characterize probability distributions
Generate moments of distribution through differentiation
Particularly useful for studying sums of independent random variables
implies existence of all moments of distribution
Prove Central Limit Theorem and other important results in probability theory
Domain always an open interval containing zero
May or may not be entire real line depending on distribution
Applications in Probability Theory
Calculate nth moment by evaluating nth derivative of MGF at t = 0
MGF of sum of independent random variables equals product of individual MGFs
Some distributions lack MGFs ()
Used to analyze linear combinations of independent random variables (a + bX, where a and b are constants)
Facilitate computation of skewness and kurtosis using higher-order derivatives
Deriving Moment Generating Functions
Discrete Distributions
Calculate MGF for discrete distributions using formula:
MX(t)=∑etxP(X=x)
Sum taken over all possible values of X
Bernoulli distribution MGF with parameter p:
MX(t)=q+pet, where q = 1 - p
MGF with parameter λ:
MX(t)=exp(λ(et−1))
Continuous Distributions
Calculate MGF for continuous distributions using formula:
MX(t)=∫etxf(x)dx
f(x) represents probability density function
Integral taken over support of X
MGF with parameter λ:
MX(t)=λ−tλ for t < λ
MGF with mean μ and variance σ^2:
MX(t)=exp(μt+2σ2t2)
MGF with shape parameter α and rate parameter β:
MX(t)=(1−βt)−α for t < β
Applying Moment Generating Functions for Moments
Calculating Basic Moments
First moment (mean) calculated by evaluating first derivative of MGF at t = 0:
E[X]=[MX′(0)](https://www.fiveableKeyTerm:mx′(0))
Second moment calculated using second derivative:
E[X2]=MX′′(0)
Variance computed using first and second moments:
Var(X)=E[X2]−(E[X])2=MX′′(0)−(MX′(0))2
Examples:
For normal distribution, E[X]=μ and Var(X)=σ2
For Poisson distribution, E[X]=Var(X)=λ
Higher-Order Moments and Statistical Measures
Calculate higher-order moments using higher-order derivatives of MGF
Skewness calculated using third standardized moment
Involves third derivative of MGF
Kurtosis calculated using fourth standardized moment
Involves fourth derivative of MGF
Central moments computed using combinations of lower-order moments