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As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random ...
The factorial moment generating function generates the factorial moments of the probability distribution. Provided M X {\displaystyle M_{X}} exists in a neighbourhood of t = 1, the n th factorial moment is given by [ 1 ]
The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality). Hoeffding's inequality. Suppose X 1, ..., X n are independent random variables taking values in [a,b]. Let X denote their sum and let μ = E[X] denote the sum's expected value.
For a degenerate point mass at c, the cumulant generating function is the straight line () =, and more generally, + = + if and only if X and Y are independent and their cumulant generating functions exist; (subindependence and the existence of second moments sufficing to imply independence.
The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing. For k = 1, the density function tends to 1/λ as x approaches zero from above and is
If P and Q are probability distributions on the real line, such that P is absolutely continuous with respect to Q, i.e. P << Q, and whose first moments exist, then (′ ()), where is the rate function, i.e. the convex conjugate of the cumulant-generating function, of , and ′ is the first moment of .
In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable.Factorial moments are useful for studying non-negative integer-valued random variables, [1] and arise in the use of probability-generating functions to derive the moments of discrete random variables.
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.