Moment-generating function

A probability distribution’s moment-generating function is the expectation of the exponential of a scaled random variable. When it exists near zero, its derivatives at zero give the distribution’s moments.

Expected value: The probability-weighted average value of a random variable. Taking the expectation of the exponential defines the moment-generating function.
Sum of independent random variables: The sum formed from random variables that are mutually independent. Independence makes the moment-generating function of a sum equal to a product.
Probability distribution: A mathematical description assigning probabilities to the possible values of a random quantity. The function summarizes a distribution through an expectation.
Probability-generating function: A power series whose coefficients are the probabilities of the nonnegative integer values of a random variable. It encodes discrete probabilities directly, while this function encodes moments through derivatives.
Taylor series: A representation of a function as a power series built from its derivatives at a point. Expanding the exponential shows how moments enter as coefficients.
Chernoff bound: An exponential upper bound on a random variable’s tail probability derived using its exponential moments. Evaluating the function at a positive argument yields bounds on upper-tail probabilities.
Random variable: A measurable quantity whose value is determined by the outcome of a random experiment. The exponential inside the defining expectation is applied to a random variable.
Laplace transform: An integral transform that maps a function or measure to an exponential-weighted integral. It is closely related by a change of sign in the exponential parameter.
Raw moment: The expectation of an integer power of a random variable, such as its mean or second moment. Each derivative at zero yields the corresponding raw moment when differentiation is valid.
Concentration inequality: A bound showing that a random quantity is unlikely to deviate far from a typical value. Moment-generating functions underpin many exponential concentration bounds.

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