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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
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
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
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)
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
Measure of the shape of a function
moment L-moment Method of moments (probability theory) Method of moments (statistics) Moment-generating function Moment measure Second moment method Standardised
Moment_(mathematics)
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
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
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
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
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
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
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
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
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
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
{\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
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 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
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 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
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
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
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
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
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
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
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
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
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
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 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
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
^{(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
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
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
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
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
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
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
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
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
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
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
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
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
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Maxwell's theorem Moment-generating function Factorial moment generating function Negative probability Probability-generating function Vysochanskiï–Petunin
List_of_probability_topics
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
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
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)
every measurable set A, where M Q {\displaystyle M_{Q}} is the moment-generating function of Q. (Note that Q0 = Q.) Then D K L ( P ∥ Q ) = D K L ( P ∥ Q
Kullback's_inequality
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 {\displaystyle
Sub-Gaussian_distribution
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
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
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
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
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
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
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
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
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
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
Concept in probability theory
{R} ^{d}} . Its branching mechanism is defined by its factorial moment generating function (the definition of a branching mechanism varies slightly among
Superprocess
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
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
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
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
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
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
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
Random matrix with gaussian entries
{64}{9\pi }}s^{2}\right)&\beta =4\\\end{cases}}} For GOE(N), its moment generating function is E [ e Tr ( V W N ) ] = e 1 4 ‖ V + V T ‖ F 2 {\textstyle
Gaussian_ensemble
Generalization of gamma distribution to multiple dimensions
_{ij}^{2})^{1/2},0)} . It is also possible to write down the moment-generating function even in the noncentral case (essentially the nth power of Craig
Wishart_distribution
Family of continuous probability distributions
needed] The following moments can be easily computed using the moment generating function of the sufficient statistic: E ( ln T ) = ψ ( α ) − ln β
Normal-gamma_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
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
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
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
Discrete probability distribution
j-m}} Each raw moment and each central moment can be easily determined with the moment generating function, but the formulas involved are
Hardy_distribution
in his Théorie analytique des probabilités (1812), introducing moment-generating function, method of least squares, inductive probability, and hypothesis
History_of_probability
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