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The 19 pages that link to Moment-generating function, each with the reason it gives.

• Probability distributionRelated: It can characterize a distribution and make sums of independent variables easier to analyze.

• Characteristic functionCompared with: Unlike this function, it may fail to exist for heavy-tailed distributions.

• Cumulative distribution functionCompared with: It summarizes moments and can identify distributions, but is not generally defined everywhere.

• Independent and identically distributed random variablesRelated: Independence turns the moment-generating function of a sum into a product of identical functions.

• Bernoulli distributionRelated: For a Bernoulli variable it is 1−p+peᵗ, yielding its moments.

• Probability-generating functionCompared with: It encodes ordinary moments, whereas this function's coefficients encode discrete probabilities.

• Chernoff boundRelated: Its values control the exponential tail estimate.

• Large deviation theoryRelated: Its exponential moments provide bounds and often determine the rate function.

• Concentration inequalityRelated: Exponential-moment methods turn moment information into tail bounds.

• Gamma distributionRelated: The gamma distribution’s moment-generating function yields its moments and sum behavior.

• Hoeffding's inequalityRelated: The proof controls exponential moments of the centered sum.

• Chi-squared distributionRelated: For this distribution, it gives a compact route to its moments and sums of independent variables.

• Log-normal distributionRelated: The log-normal distribution has no finite moment-generating function for positive arguments.

• Hermite–Hadamard inequalityRelated: Convexity estimates can bound exponential averages and hence moment-generating functions.

• Leibniz integral ruleRelated: Its defining expectation is differentiated through an integral to obtain moments.

• KurtosisRelated: Its fourth derivative can yield the fourth moment used to calculate kurtosis.

• Faà di Bruno's formulaRelated: Differentiating its exponential representation invokes the formula's partition structure.

• Cochran's theoremRelated: Transform arguments can establish the chi-squared laws and independence asserted by the theorem.

• Cramér's decomposition theoremCompared with: Unlike characteristic functions, moment-generating functions need not exist for every summand.