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Exploring Data Measures of Centrality and Dispersion	Graphical Procedures Histograms, Boxplots, Ogives	Probability and Random Variables	Probability Distributions Discrete and Continuous probability Distributions. Compound and Mixed Probability Distributions
Testing Distributional Assumptions Testing Normality and Outlier Detection	Inference Procedures Confidence Intervals and Hypothesis Testing	Bivariate Data Correlation Coefficients & Hypothesis Tests, Chi Square Test for Independence	Linear Models Simple and Multiple Linear Regression Models, Residual Analysis, Regression ANOVA
Discrete Random Variables Expectations and variance of discrete random variables, Joint Distribution of discrete random variables	Continuous Random Variables Expectations and variance of continuous random variables	More on Random Variables Moment Generating Functions, Cumulant Generating Functions	Random Number Generation
Game Theory and Decision Theory Matrix Games	Experimental Design ANOVA Procedures, Testing Model Assumptions	Survival Analysis Non parametric approaches, Cox proportional hazard models	Statistical Process Control
Time Series Analysis Time Series

Moment Generating Functions

Moment Generating Functions

• Cumulant Generating Functions
  
• Moment Generating Functions
  
• Moments of Discrete Random Variable (Short Definitions)

MGFs for Discrete Distributions

• Binomial Distribution - Probability and Moment Generating Functions

Cumulant Generating Function

The cumulants of a random variable \(\LARGE X\) are defined using the cumulant-generating function \(C(t)\), which is the natural logarithm of the moment-generating function:

\[ {\LARGE C(t)=\log \operatorname {E} \left[e^{tX}\right].}\]

The cumulants \(\kappa_n\) are obtained from a power series expansion of the cumulant generating function: \[ {\LARGE C(t)=\sum _{n=1}^{\infty }\kappa _{n}{\frac {t^{n}}{n!}}=\mu t+\sigma ^{2}{\frac {t^{2}}{2}}+\cdots .}\]

This expansion is a Maclaurin series, so the \(n\)-th cumulant can be obtained by differentiating the above expansion \(n\) times and evaluating the result at zero:

\[{\LARGE \kappa _{n}=K^{(n)}(0).} \]

Worked Examples

The cumulant generating function of a random variable X is given by: \[C_X (t) = \log M_X (t) = 2\left[ (1\;-\;t)^{-10} \;-\;1\right]\] where \(M_X(t)\) is the moment generating function.

Exercises

Determine the mean and variance of the distribution of \(\LARGE X\).

Solution

Click here for demonstrated solution