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Concept in probability theory and statistics
derivative of the moment generating function, evaluated at 0. In addition to univariate real-valued distributions, moment generating functions can also be defined
Moment_generating_function
Formal power series
generating functions of note include the entries in the next table, which is by no means complete. Moment-generating function Probability-generating function
Generating_function
Power series derived from a discrete probability distribution
the probability generating function (of X {\displaystyle X} ) and M X ( t ) {\displaystyle M_{X}(t)} is the moment-generating function (of X {\displaystyle
Probability generating function
Probability_generating_function
Set of quantities in probability theory
the cumulant generating function (CGF) K(t), which is a generating function that is the natural logarithm of the moment generating function: K ( t ) = log
Cumulant
In mathematics, a quantitative measure of the shape of a set of points
moment L-moment Method of moments (probability theory) Method of moments (statistics) Moment-generating function Moment measure Second moment method Standardised
Moment_(mathematics)
Fourier transform of the probability density function
a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Compound probability distribution
π {\displaystyle M_{\pi }} is the moment generating function of the density. For the probability generating function, one obtains m X ( s ) = M π ( s −
Mixed_Poisson_distribution
Continuous probability distribution
{\displaystyle {\text{MTBF}}(k,\lambda )=\lambda \Gamma (1+1/k).} The moment generating function of the logarithm of a Weibull distributed random variable is given
Weibull_distribution
Probability distribution
\operatorname {E} [X^{k}]} . The cumulant generating function is the logarithm of the moment generating function, namely g ( t ) = ln M ( t ) = μ t + 1
Normal_distribution
Probability distribution
determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value E [ e
Log-normal_distribution
Uniform distribution on an interval
height would be 1 15 . {\displaystyle {\tfrac {1}{15}}.} The moment-generating function of the continuous uniform distribution is: M X = E [ e t X ]
Continuous uniform distribution
Continuous_uniform_distribution
Stochastic process for effort or wear
where Γ ( z ) {\displaystyle \Gamma (z)} is the Gamma function. The moment generating function is the expected value of exp ( t X ) {\displaystyle \exp(tX)}
Gamma_process
Probability distribution
confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated as M ( t
Wigner semicircle distribution
Wigner_semicircle_distribution
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable
Factorial moment generating function
Factorial_moment_generating_function
Exponentially decreasing bounds on tail distributions of random variables
decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff
Chernoff_bound
{\mu }}^{T})} includes parameters of the distribution. The joint moment generating function of G-MVLG distribution is as the following: M Y ( t ) = δ ν (
Generalized multivariate log-gamma distribution
Generalized_multivariate_log-gamma_distribution
Noncentral generalization of the chi-squared distribution
in the series are (1 + 2i) + (k − 1) = k + 2i as required. The moment-generating function is given by M ( t ; k , λ ) = exp ( λ t 1 − 2 t ) ( 1 − 2 t
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Theorem In probability theory and statistics
by Harry Bateman. In Campbell's work, he presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks
Campbell's theorem (probability)
Campbell's_theorem_(probability)
Probability distribution in mathematics
the series itself, and are therefore undefined for large n. The moment generating function is defined as M ( t ; s ) = E ( e t X ) = 1 ζ ( s ) ∑ k = 1 ∞
Zeta_distribution
Probability distribution
\end{aligned}}} In particular MX(α; β; 0) = 1. Using the moment generating function, the k-th raw moment is given by the factor ∏ r = 0 k − 1 α + r α + β +
Beta_distribution
Statistical approximation method
formula for any PDF or probability mass function of a distribution, based on the moment generating function. There is also a formula for the CDF of the
Saddlepoint approximation method
Saddlepoint_approximation_method
Discrete-variable probability distribution
and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a
Probability_mass_function
Probability distribution
fractional absolute moments exist. The Cauchy distribution has no moment generating function. In mathematics, it is closely related to the Poisson kernel,
Cauchy_distribution
Family of probability distributions related to the normal distribution
form for the moment-generating function for the distribution of x. In particular, using the properties of the cumulant generating function, E ( T j )
Exponential_family
Moment of a random variable minus its mean
univariate probability distribution with probability density function f(x), the n-th moment about the mean μ is μ n = E [ ( X − E [ X ] ) n ] = ∫ −
Central_moment
Continuous probability distribution, named after Benjamin Gompertz
{\displaystyle \eta ,b>0,} and x ≥ 0 . {\displaystyle x\geq 0\,.} The moment generating function is: E ( e − t X ) = η e η E t / b ( η ) {\displaystyle
Gompertz_distribution
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Statistical probability Distribution for discrete event counts
e^{t}-1)+a_{2}(e^{2t}-1))} The cumulant generating function is the logarithm of the moment generating function and is equal to K ( t ) = log ( M ( t
Hermite_distribution
Fundamental result in the theory of large deviations
sequence of iid real random variables with finite logarithmic moment generating function, i.e. Λ ( t ) < ∞ {\displaystyle \Lambda (t)<\infty } for all
Cramér's theorem (large deviations)
Cramér's_theorem_(large_deviations)
Probability distribution
distribution do not exist (only some fractional moments). The moment-generating function would be formally defined by M ( t ; c ) = d e f c 2 π ∫ 0
Lévy_distribution
Family of probability distributions
has a moment generating function M X ( t ) {\displaystyle M_{X}(t)} , then Y = a + b X {\displaystyle Y=a+bX} has a moment generating function M Y ( t
Location–scale_family
Probability distribution
special and limiting cases. Using similar notation as above, the moment-generating function of the EGB can be expressed as follows: M E G B ( Z ) = e δ t
Generalized_beta_distribution
Average value of a random variable
variables can be used to specify their distributions, via their moment generating functions. To empirically estimate the expected value of a random variable
Expected_value
Measure of the deviation of position over time
the moment-generating function, an extremely useful, and general function when dealing with probability densities. The moment-generating function describes
Mean_squared_displacement
Probability distribution
from E [ X ] c {\displaystyle \operatorname {E} [X]^{c}} . The moment-generating function is M X ( t ) = E [ e t X ] = ( 1 − p + p e t ) n {\displaystyle
Binomial_distribution
Probability distribution
= 0 {\displaystyle t=0} we say that the moment generating function does not exist. The characteristic function is given by φ ( t ; α , x m ) = α ( − i
Pareto_distribution
Inequality in probability theory
probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable, implying that such variables are
Hoeffding's_lemma
Probability distribution
\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}} The moment generating function is given by: E ( e − t x ) = { β s s b t + s b 2 F 1 ( s + 1
Gamma/Gompertz_distribution
Probability distribution
^{(1)}} is the trigamma function. This can be derived using the exponential family formula for the moment generating function of the sufficient statistic
Gamma_distribution
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Integral transform useful in probability theory, physics, and engineering
of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result. Laplace's use of generating functions was
Laplace_transform
Continuous probability distribution
continuous probability distribution defined by a unique convex quadratic function with lower limit a and upper limit b. f ( x | a , b , α , β ) = α ( x −
U-quadratic_distribution
Mathematical transformation
deviations theory, the rate function is defined as the Legendre transformation of the logarithm of the moment generating function of a random variable. An
Legendre_transformation
Mathematical function for the probability a given outcome occurs in an experiment
probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and
Probability_distribution
Probability distribution
moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either
Voigt_profile
Random process in probability theory
\end{aligned}}} Lastly, using the law of total probability, the moment generating function can be given as follows: Pr ( Y ( t ) = i ) = ∑ n Pr ( Y ( t )
Compound_Poisson_process
Probability problem
distribution function and the probability density function can often be found by applying the inverse Laplace transform to the moment generating function m ( t
Hamburger_moment_problem
Maxwell's theorem Moment-generating function Factorial moment generating function Negative probability Probability-generating function Vysochanskiï–Petunin
List_of_probability_topics
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
characteristic function is φ ( k ) = sin ( k / 2 ) k / 2 , {\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}},} and its moment-generating function is M ( k
Rectangular_function
Differentiation under the integral sign formula
such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments
Leibniz_integral_rule
Compound Poisson-family discrete probability distribution
(e^{\phi (e^{t}-1)}-1))} The cumulant generating function is the logarithm of the moment generating function and is equal to K ( t ) = log ( M ( t
Neyman_Type_A_distribution
Generalized function whose value is zero everywhere except at zero
characteristic function and moment generating function are both equal to one. In the theory of distributions, a generalized function is considered not a function in
Dirac_delta_function
Measure of the asymmetry of random variables
central moment, and κt are the t-th cumulants. It is sometimes referred to as Pearson's moment coefficient of skewness, or simply the moment coefficient
Skewness
Type of function space
easily computed from a strictly monotonic moment-generating function. For example, the moment-generating function of a chi-squared random variable X with
Orlicz_space
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
Probability distribution
where M g {\displaystyle M_{g}} is the moment generating function of the probability distribution with density function g {\displaystyle g} , i.e. M g ( s
Normal_variance-mean_mixture
Mathematical operation
transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier
Two-sided_Laplace_transform
Class of probability distributions
\mathbb {R} ^{p}.} A member of a natural exponential family has moment generating function (MGF) of the form M X ( t ) = exp ( A ( θ + t ) − A ( θ )
Natural_exponential_family
Probability distribution
, x ) {\displaystyle P(k,x)} is the regularized gamma function. The moment-generating function is given by: M ( t ) = M ( k 2 , 1 2 , t 2 2 ) + t 2 Γ
Chi_distribution
Model in probability theory
independent and identically distributed random variables with moment-generating function M ( θ ) = E [ exp ( θ X 1 ) ] {\displaystyle M(\theta )=\mathbf
Martingale (probability theory)
Martingale_(probability_theory)
Variable representing a random phenomenon
identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform
Random_variable
Statistical measure of how far values spread from their average
the square root of the variance. Technically, it is the second central moment of a distribution, and the covariance of the random variable with itself
Variance
Coherent measure for value at risk
set of all Borel measurable functions X : Ω → R {\displaystyle X:\Omega \to \mathbb {R} } whose moment-generating function M X ( z ) {\displaystyle M_{X}(z)}
Entropic_value_at_risk
Estimated potential loss for an investment under a given set of conditions
\mathbf {L} _{M^{+}}} the set of all Borel measurable functions whose moment-generating function exists for all positive real values) we have VaR 1 − α
Value_at_risk
Probability distribution and special case of gamma distribution
Expectation of the log moment of gamma. For derivation from more basic principles, see the derivation in moment-generating function of the sufficient statistic
Chi-squared_distribution
Topics referred to by the same term
muscles in response to training, considered an isoform of IGF-1 Moment-generating function, in probability and statistics .mgf, (for Mascot generic format)
MGF
Continuous probability distribution
distributions. The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available
Variance-gamma_distribution
Analytic function in mathematics
The Brownian motion and Riemann zeta function are connected through the moment-generating functions of stochastic processes derived from the Brownian
Riemann_zeta_function
Probability distribution
1 / 6 1 / 6 = 5 {\displaystyle {\frac {1-1/6}{1/6}}=5} . The moment generating function of the geometric distribution when defined over N {\displaystyle
Geometric_distribution
Name for several different families of probability distributions
distribution. The cumulant generating function is K ( t ) = ln M ( t ) {\displaystyle K(t)=\ln M(t)} , where the moment generating function M ( t ) {\displaystyle
Generalized logistic distribution
Generalized_logistic_distribution
Logmoment generating function Marcinkiewicz–Zygmund inequality / inq Method of moments / lmt (L:R) Moment problem / anl (1:R) Moment-generating function / anl
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Type of probability distribution
[X])t}]\leq e^{\frac {K^{2}t^{2}}{2}}} for all t {\displaystyle t} ; Moment-generating function (of X 2 {\displaystyle X^{2}} ): E [ e X 2 t 2 ] ≤ e K 2 t 2
Sub-Gaussian_distribution
Probability distribution
{\displaystyle \operatorname {erfi} (z)} is the imaginary error function. The moment generating function is given by M ( t ) = 1 + σ t e 1 2 σ 2 t 2 π 2 [ erf
Rayleigh_distribution
Stochastic process
modeling default times in credit risk applications, since both the moment generating function m ( q ) = E ( e q ∫ 0 t Z s d s ) , q ∈ R , {\displaystyle
Basic_affine_jump_diffusion
Statistical sequence characterizing probability distributions
identical to the conventional mean). Standardized L-moments are called L-moment ratios and are analogous to standardized moments. Just as for conventional
L-moment
Continuous probability distribution
NIG-triangle. The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available
Normal-inverse Gaussian distribution
Normal-inverse_Gaussian_distribution
Theorem in mathematics
equations yield the Dirac comb identity. Moment-generating function of a random variable An example is the MATLAB function, hilbert(u,N). McGillem, Clare D.;
Convolution_theorem
variable's cumulative distribution function is therefore equal to the random variable's moment-generating function, but with the sign of the argument
Laplace–Stieltjes_transform
distribution, D, is a sub-distribution of another distribution E if D 's moment-generating function, is bounded by E 's up to a constant. In other words E X ∼ D [
Super-Poissonian_distribution
Discrete probability distribution
{n}{k}}^{\nu }.} Then, the probability generating function, moment generating function and characteristic function are given, respectively, by: G ( t )
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
Wigner distribution function in physics as opposed to in signal processing
probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x)
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Y=Xe^{hX}/m_{X}(h)} , where m X ( h ) {\displaystyle m_{X}(h)} denotes the moment generating function. This risk measure does not respect the positive homogeneity property
Esscher_transform
Expectation or average of the falling factorial of a random variable
numbers of the second kind. Factorial moment measure Moment (mathematics) Cumulant Factorial moment generating function The Pochhammer symbol (x)r is used
Factorial_moment
Discrete probability distribution
moments in a way that no Skellam distribution can satisfy. The moment-generating function is given by: M ( t ; μ 1 , μ 2 ) = G ( e t ; μ 1 , μ 2 ) = ∑ k
Skellam_distribution
Continuous probability distribution
equal), so the coefficient of variation is greater than 1. The moment-generating function is given by E [ e t x ] = ∫ − ∞ ∞ e t x f ( x ) d x = ∑ i = 1
Hyperexponential_distribution
Probability distribution
random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, MX(t), is infinite for all t > 0
Heavy-tailed_distribution
Probability distribution
for the Dirichlet distribution. Another inequality relates the moment-generating function of the Dirichlet distribution to the convex conjugate of the scaled
Dirichlet_distribution
upper bound. (Also written sup.) max – maximum of a set. MGF – moment-generating function. M.I. – mathematical induction. min – minimum of a set. mod –
List of mathematical abbreviations
List_of_mathematical_abbreviations
in his Théorie analytique des probabilités (1812), introducing moment-generating function, method of least squares, inductive probability, and hypothesis
History_of_probability
Mathematical theory
variable X are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as G X ( t ) = M X ( log ( 1
Combinant
Probability distribution
Poisson binomial distribution gets large, can be bounded using its moment generating function as follows (valid when s ≥ μ {\displaystyle s\geq \mu } and for
Poisson_binomial_distribution
Moffat distribution Moment (mathematics) Moment-generating function Moments, method of – see method of moments (statistics) Moment problem Monotone likelihood
List_of_statistics_articles
Mathematical inequality explaining concentration of random variables
deviation probability. The generic Chernoff bound requires the moment generating function of X {\displaystyle X} , defined as M X ( t ) := E [ e t X ]
Concentration_inequality
Overview of and topical guide to probability
Probability-generating functions Moment-generating functions Laplace transforms and Laplace–Stieltjes transforms Characteristic functions A proof of the
Outline_of_probability
Statistical hypothesis test
just like a conventional fourth-order Edgeworth expansion. The moment generating function of T {\displaystyle T} has the exact formula: M ( t ) = 1 2 n
Wilcoxon_signed-rank_test
Family of probability distributions often used to model tails or extreme values
>0} and − ∞ < ξ < ∞ {\displaystyle -\infty <\xi <\infty } . The moment-generating function of Y ∼ e x G P D ( σ , ξ ) {\displaystyle Y\sim \mathrm {exGPD}
Generalized Pareto distribution
Generalized_Pareto_distribution
Family of continuous probability distributions
cumulant generating functions of the Gaussian and inverse Gaussian distributions are inverse of each other (i.e., the graphs of the two cumulant generating functions
Inverse_Gaussian_distribution
Aspect of queueing theory
)(1-s)}{1-s/M_{S}(\lambda (s-1))}}} where M S {\displaystyle M_{S}} is the moment-generating function of a general service time. The stationary distribution of an M/G/1
M/G/1_queue
Phase of a cycle
interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents. The geometric phase can be evaluated exactly
Geometric_phase
Three-parameter family of continuous probability distributions
distribution. The modulus of z, |z|, then has K-distribution. The moment generating function is given by M X ( s ) = ( ξ s ) β / 2 exp ( ξ 2 s ) W − δ /
K-distribution
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
Girl/Female
Tamil
Moment
Girl/Female
Hindu
Moment
Boy/Male
Australian, British, English, French
Moment
Surname or Lastname
English
English : topographic name for someone who lived on or near a hill, Middle English mount (from Old English munt, reinforced by Old French mont).Scottish : probably a habitational name from places so called in Peeblesshire, Fife, and Lanarkshire.
Girl/Female
Indian
Omen, Luck, Fortunate, Auspicious moment
Boy/Male
Indian, Modern
Generations
Surname or Lastname
English (Lincolnshire and Yorkshire)
English (Lincolnshire and Yorkshire) : unexplained.
Girl/Female
French, German, Latin, Spanish
Modest
Surname or Lastname
English
English : variant of Diamond 2.
Girl/Female
Gujarati, Haryanvi, Hebrew, Hindu, Indian, Kannada, Punjabi, Sikh
Moment of Life; Every Movement; God Time
Girl/Female
Muslim
Moment
Girl/Female
Muslim
Omen, Luck, Fortunate, Auspicious moment
Girl/Female
Tamil
Moment
Boy/Male
Muslim
Believer and faithful to Allah
Girl/Female
Indian, Tamil
Generation
Boy/Male
Hindu, Indian, Sanskrit
Moment
Girl/Female
Indian
Generation
Male
Slovene
Slovene form of Latin Dominicus, DOMEN means "belongs to the lord."
Girl/Female
Hindu
Moment
Male
Russian
(МодеÑÑ‚) Russian form of Roman Latin Modestus, MODEST means "moderate, sober."
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
Surname or Lastname
English
English : variant of Pettengill.
Boy/Male
Hindu
The universe
Male
French
French form of Latin Desiderius, DIDIER means "longing."Â
Female
Greek
(Λητώ) Greek name LÊTÔ means "the hidden one." In mythology, this is the name of the mother of Apollo and Artemis.
Boy/Male
Indian
Autumn in Chinese
Male
Norse
Old Norse name derived from ancient *wihaR, "battle, fight," hence "fighter, warrior."
Girl/Female
Hindu, Indian, Sanskrit
Pure
Boy/Male
English
River ford near a cliff.
Girl/Female
English German
Winged.
Girl/Female
Arabic, Muslim
Healthy
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
MOMENT GENERATING-FUNCTION
a.
Generating bile.
n.
Impulsive power; force; momentum.
a.
Acute; discerning; sagacious; quick to discover; as, a penetrating mind.
adv.
For a moment.
pl.
of Momentum
a.
Lasting but a moment; brief.
n.
Origination by some process, mathematical, chemical, or vital; production; formation; as, the generation of sounds, of gases, of curves, etc.
a.
Powerful, in an intellectual or moral sense; having great influence; as, potent interest; a potent argument.
adv.
In a moment; every moment; momentarily.
a.
Pertaining to generation, or to the generative organs.
v. t.
To comment on.
n.
The act of generating or begetting; procreation, as of animals.
n.
A minute portion of time; a point of time; an instant; as, at thet very moment.
v. t.
To nurse to life or activity; to cherish and promote by excitements; to encourage; to abet; to instigate; -- used often in a bad sense; as, to foment ill humors.
n.
To overlay or coat with cement; as, to cement a cellar bottom.
a.
Having the power of entering, piercing, or pervading; sharp; subtile; penetrative; as, a penetrating odor.
a.
Having the power of generating, propagating, originating, or producing.
a.
Of or pertaining to moment or momentum.
a.
Generating mucus.
adv.
Every moment; from moment to moment.