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Power series derived from a discrete probability distribution
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of
Probability generating function
Probability_generating_function
Concept in probability theory and statistics
In probability theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative
Moment_generating_function
Fourier transform of the probability density function
include the moment-generating function and the probability-generating function. The characteristic function exists for all probability distributions. This
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Description of continuous random distribution
In probability theory, a probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function
Probability_density_function
Formal power series
is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function. The exponential
Generating_function
Discrete-variable probability distribution
In probability and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the
Probability_mass_function
Probability that random variable X is less than or equal to x
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution
Cumulative distribution function
Cumulative_distribution_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
Statistical function that defines the quantiles of a probability distribution
In probability and statistics, the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the
Quantile_function
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
Probability_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
Generalization of the concept from statistical mechanics
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition
Partition function (mathematics)
Partition_function_(mathematics)
Compound Poisson-family discrete probability distribution
develops, we must bear in mind that the probability mass function is calculated from the probability generating function, and use the property of Stirling Numbers
Neyman_Type_A_distribution
French polymath (1749–1827)
to a different variable. The latter is therefore called the probability-generating function of the former. Laplace then shows how, by means of interpolation
Pierre-Simon_Laplace
Maxwell's theorem Moment-generating function Factorial moment generating function Negative probability Probability-generating function Vysochanskiï–Petunin
List_of_probability_topics
Moment of a random variable minus its mean
expectation operator. For a continuous univariate probability distribution with probability density function f(x), the n-th moment about the mean μ is μ n
Central_moment
Discrete probability distribution
} One derivation of this uses probability-generating functions. Consider a Bernoulli trial (coin-flip) whose probability of one success (or expected number
Poisson_distribution
Discrete probability distribution
_{k=-\infty }^{\infty }p(k;\mu _{1},\mu _{2})=1.} We know that the probability generating function (pgf) for a Poisson distribution is: G ( t ; μ ) = e μ ( t
Skellam_distribution
Statistical model allowing for frequent zero values
{\displaystyle G(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}} be the probability generating function of y i {\displaystyle y_{i}} . If p 0 = Pr ( Y = 0 ) > 0.5
Zero-inflated_model
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
Moment_(mathematics)
Statistical probability Distribution for discrete event counts
called it "Hermite distribution" from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of
Hermite_distribution
Function related to statistics and probability theory
likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing
Likelihood_function
Overview of and topical guide to probability
transforms) Probability-generating functions Moment-generating functions Laplace transforms and Laplace–Stieltjes transforms Characteristic functions A proof
Outline_of_probability
Probability distribution
gamma function. Using that f ( . ; m, r, ps) for s ∈ (0, 1] is also a probability mass function, it follows that the probability generating function is given
Extended negative binomial distribution
Extended_negative_binomial_distribution
Probability distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes
Binomial_distribution
Probability distribution
this, we calculate the probability generating function GX of X, which is the composition of the probability generating functions GN and GY1. Using G N
Negative binomial distribution
Negative_binomial_distribution
Statistical measure of how far values spread from their average
generator of random variable X {\displaystyle X} is discrete with probability mass function x 1 ↦ p 1 , x 2 ↦ p 2 , … , x n ↦ p n {\displaystyle x_{1}\mapsto
Variance
Conditional probability used in Bayesian statistics
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood
Posterior_probability
Probability distribution
{1-p}{p(1-p)^{2}}}\right)\\&={\frac {1}{p^{2}(1-p)}}\end{aligned}}} The probability generating functions of geometric random variables X {\displaystyle X} and Y {\displaystyle
Geometric_distribution
Aspect of queueing theory
Policies can also be evaluated using a measure of fairness. The probability generating function of the stationary queue length distribution is given by the
M/G/1_queue
Average value of a random variable
In probability theory, the expected value (also called expectation, mean, or first moment) is a generalization of the weighted average. The expected value
Expected_value
Uniform distribution on an interval
than that it is contained in the distribution's support. The probability density function of the continuous uniform distribution is f ( x ) = { 1 b − a
Continuous uniform distribution
Continuous_uniform_distribution
Expectation or average of the falling factorial of a random variable
non-negative integer-valued random variables, and arise in the use of probability-generating functions to derive the moments of discrete random variables. Factorial
Factorial_moment
Topics referred to by the same term
graphics language in the PGF/TikZ pair Precision guided firearm Probability-generating function Progressive Graphics File, a file format This disambiguation
PGF
Linear transform from the time domain to the frequency domain
series Generating function Generating function transformation Laplace transform Laurent series Least-squares spectral analysis Probability-generating function
Z-transform
Measure of the asymmetry of random variables
Skewness in probability theory and statistics is a measure of the asymmetry of the probability distribution of a real-valued random variable about its
Skewness
Probability plot correlation coefficient plot Probability space Probability theory Probability-generating function Probable error Probit Probit model Procedural
List_of_statistics_articles
Compound probability distribution
{\displaystyle M_{\pi }} is the moment generating function of the density. For the probability generating function, one obtains m X ( s ) = M π ( s − 1
Mixed_Poisson_distribution
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over
Softmax_function
Algebraic structure of set algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In
Σ-algebra
Probability distribution
distribution for a real-valued random variable. The general form of its probability density function is f ( x ) = 1 2 π σ 2 exp ( − ( x − μ ) 2 2 σ 2 ) . {\displaystyle
Normal_distribution
Theorem about convergence of Fourier series
\Pr(X=i)=P_{i}} , i ∈ N {\displaystyle i\in \mathbb {N} } , has the probability generating function of the form P ( z ) = ∑ i = 0 ∞ P i z i = exp { ∑ i = 1 ∞
Wiener–Lévy_theorem
Type of probability space
map from the unit interval to the space of continuous functions. The theory of standard probability spaces was started by von Neumann in 1932 and shaped
Standard_probability_space
Type of random mathematical object
point process. The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined
Poisson_point_process
Kind of mathematical function
in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Let ( X
Measurable_function
Distribution function associated with the empirical measure of a sample
distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that
Empirical distribution function
Empirical_distribution_function
Model in probability theory
\chi _{F}} denotes the indicator function of the event F {\displaystyle F} . In Grimmett and Stirzaker's Probability and Random Processes, this last condition
Martingale (probability theory)
Martingale_(probability_theory)
Kind of stochastic process
right-hand side of the equation is a probability generating function. Let h(z) be the ordinary generating function for pi: h ( z ) = p 0 + p 1 z + p 2
Branching_process
Variance of random sum
derivation can be done elementarily using the chain rule and the probability-generating function. For each n ≥ 0 {\displaystyle n\geq 0} , let χ n {\displaystyle
Blackwell-Girshick_equation
Probability function
large deviations theory, a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation
Rate_function
Mathematical theory
mathematical theory of probability, the combinants cn of a random variable X are defined via the combinant-generating function G(t), which is defined
Combinant
Mathematical concept
in the sample space. A probability function, P {\displaystyle P} , which assigns, to each event in the event space, a probability, which is a number between
Probability_space
Interpretation of probability
The propensity theory of probability is a probability interpretation in which the probability is thought of as a physical propensity, disposition, or tendency
Propensity_probability
The Dirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents
List of probability distributions
List_of_probability_distributions
Class of distance functions defined between probability distributions
In probability theory, integral probability metrics are types of distance functions between probability distributions, defined by how well a class of functions
Integral_probability_metric
Basic method for pseudo-random number sampling
sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation
Inverse_transform_sampling
Mapping arbitrary data to fixed-size values
minimize duplication of output values (collisions). Hash functions rely on generating favorable probability distributions for their effectiveness, reducing access
Hash_function
Statistical approximation method
approximation formula for any PDF or probability mass function of a distribution, based on the moment generating function. There is also a formula for the
Saddlepoint approximation method
Saddlepoint_approximation_method
Variable representing a random phenomenon
variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides
Random_variable
Family of probability distributions
equivalent to the Tweedie compound Poisson–gamma distribution. The probability generating function for the PNB distribution is G ( s ) = exp [ λ α − 1 α ( θ
Tweedie_distribution
Average uncertainty in variable's states
very low probability event. The information content, also called the surprisal or self-information, of an event E {\displaystyle E} is a function that increases
Entropy_(information_theory)
When the occurrence of one event does not affect the likelihood of another
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically
Independence (probability theory)
Independence_(probability_theory)
Probability distribution
the normal, binomial, gamma, and Poisson distributions. The probability density function (pdf) of an exponential distribution is f ( x ; λ ) = { λ e −
Exponential_distribution
Theory of behavioral economics
It introduces a value function defined over gains and losses rather than final wealth, as well as a probability-weighting function that reflects the tendency
Prospect_theory
Philosophical interpretation of the axioms of probability
word "probability" has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure
Probability_interpretations
constructing the probability distribution of S. In the following W N ( x ) {\displaystyle W_{N}(x)\,} denotes the probability generating function of N: for this
Panjer_recursion
Monte Carlo algorithm
samples from any probability distribution with probability density P ( x ) {\displaystyle P(x)} , provided that we know a function f ( x ) {\displaystyle
Metropolis–Hastings_algorithm
Canadian mathematician
Micheal (2019). "Central limit theorems from the roots of probability generating functions". Advances in Mathematics. 358 106840. arXiv:1804.07696. doi:10
Julian_Sahasrabudhe
Probability distribution of the sum of random variables
convolution of probability distributions. Often the manipulation of integrals can be avoided by use of some type of generating function. Such methods can
Convolution of probability distributions
Convolution_of_probability_distributions
Probability distribution
to multiple variables is called a Dirichlet distribution. The probability density function (PDF) of the beta distribution, for 0 ≤ x ≤ 1 {\displaystyle
Beta_distribution
Model for the extinction of family names
The process can be treated analytically using the method of probability generating functions. If the number of children ξ j {\displaystyle \xi _{j}} at
Galton–Watson_process
Aspect of probability theory
the discrete random variable Y {\displaystyle Y} satisfying probability generating function characterization P Y ( z ) = ∑ i = 0 ∞ P ( Y = i ) z i = exp
Compound_Poisson_distribution
R is the field of real numbers, then this is the probability-generating function of the probability distribution of N. Similarly, (5) and (6) yield and
Schuette–Nesbitt_formula
Problem in probability theory
z {\displaystyle z} with 1 + z {\displaystyle 1+z} in the probability generating function produces the o.g.f. for E [ ( X k ) ] {\displaystyle E\left[{X
Coupon_collector's_problem
Probability distribution
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential
Gamma_distribution
Fundamental result in the theory of large deviations
the theory of large deviations, a subdiscipline of probability theory. It determines the rate function of a series of iid random variables. A weak version
Cramér's theorem (large deviations)
Cramér's_theorem_(large_deviations)
Mathematical function having a characteristic S-shaped curve or sigmoid curve
common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal
Sigmoid_function
Probability distribution
half-plane. It is one of the few stable distributions with a probability density function that can be expressed analytically, the others being the normal
Cauchy_distribution
Wigner distribution function in physics as opposed to in signal processing
Schrödinger equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Mathematical identity in queueing theory
{\text{Var}}(S)}{2(\mu -\lambda )}}.} Writing π(z) for the probability-generating function of the number of customers in the queue π ( z ) = ( 1 − z )
Pollaczek–Khinchine_formula
Statistical function that converts a probability to a standard normal score
In statistics, the probit function converts a probability (a number between 0 and 1) into a score. This score indicates how many standard deviations a
Probit
Mathematical function
controls the width of the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed random variable
Gaussian_function
Mathematical theory on random variables
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue
Free_probability
Mathematical rule for inverting probabilities
used to invert the probability of observations given a model configuration (i.e., the likelihood function) to obtain the probability of the model configuration
Bayes'_theorem
Probabilistic optimization technique and metaheuristic
{\displaystyle s_{\mathrm {new} }} is specified by an acceptance probability function P ( e , e n e w , T ) {\displaystyle P(e,e_{\mathrm {new} },T)}
Simulated_annealing
Probability distribution
ideal gas (chi distribution with three degrees of freedom). The probability density function (pdf) of the chi-distribution is f ( x ; k ) = { x k − 1 e −
Chi_distribution
Continuous probability distribution
cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in
Weibull_distribution
Probability distribution
physicist Eugene Wigner, is the probability distribution defined on the domain [−R, R] whose probability density function f is a scaled semicircle, i.e
Wigner semicircle distribution
Wigner_semicircle_distribution
Exponentially decreasing bounds on tail distributions of random variables
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function
Chernoff_bound
Statistical model for a binary dependent variable
probability of the value labeled "1" can vary between 0 (certainly the value "0") and 1 (certainly the value "1"), hence the labeling; the function that
Logistic_regression
Term in statistical hypothesis testing
{\displaystyle H_{1}} defines its own probability distribution for t (the difference between the two distributions being a function of the effect size), the power
Power_(statistics)
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
the basis for a rectangular wave. The rect function has been introduced 1953 by Woodward in "Probability and Information Theory, with Applications to
Rectangular_function
Halting probability of a random computer program
definition of a halting probability relies on the existence of a prefix-free universal computable function. Such a function, intuitively, represents
Chaitin's_constant
Process of using data analysis for predicting population data from sample data
the process generating the data are much less than in parametric statistics and may be minimal. For example, every continuous probability distribution
Statistical_inference
Binary tree selected at random
{\displaystyle g(r)=r} , where g {\displaystyle g} is the probability-generating function of the distribution on the number of children, here g ( x )
Random_binary_tree
Discrete probability distribution
_{k=0}^{n}x^{k}{\binom {n}{k}}^{\nu }.} Then, the probability generating function, moment generating function and characteristic function are given, respectively, by: G
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
Probability distribution
moment-generating function is actually undefined. Like all stable distributions except the normal distribution, the wing of the probability density function
Lévy_distribution
Concept in probability theory
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It
Law_of_total_probability
Model for generating observable data in probability and statistics
the conditional probability P ( Y ∣ X ) {\displaystyle P(Y\mid X)} can also be interpreted as a (non-deterministic) target function f : X → Y {\displaystyle
Generative_model
Probability distribution
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 GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
Boy/Male
Biblical
Nativity, generation.
Girl/Female
Indian
Generation
Girl/Female
Biblical
Birth, generation.
Boy/Male
Japanese Welsh
Large; generation.
Girl/Female
Biblical
Generation, habitation.
Boy/Male
Hindu, Indian
Young Generation
Girl/Female
Biblical
Nativity, generation.
Boy/Male
British, Czech, Hindu, Indian
New Generation
Boy/Male
Indian
Young Generation
Boy/Male
Tamil
Forthcoming generation
Boy/Male
Tamil
Young generation
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi
Generation; Coming Generation of Father; Family
Boy/Male
Biblical
Nativity, generation.
Boy/Male
Biblical, British, English
Nativity; Generation
Boy/Male
Gujarati, Hindu, Indian, Kannada
Era; Generation
Boy/Male
Muslim
Old generation
Girl/Female
Biblical
A generation.
Boy/Male
Indian, Punjabi, Sikh
New Generation
Boy/Male
Indian, Modern
Generations
Girl/Female
Indian, Tamil
Generation
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
Boy/Male
Hindu, Indian, Sanskrit
Right; Correct
Boy/Male
Hindu
Name of a sage
Girl/Female
Muslim/Islamic
A flower in the desert
Girl/Female
Tamil
Garland of Rudraksh
Girl/Female
Indian
One who guards her self, Flower
Girl/Female
Australian, British, English, French
The Country India; Land of the Indians
Boy/Male
Hindu, Indian, Marathi
Leading to Victory
Girl/Female
Polish
Gift of God.
Boy/Male
Hindu
Master of mount Kailash, Lord Shiva
Male
Native American
Native American Sioux name CHAYTON means "falcon."
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
a.
Generating mucus.
n.
Origination by some process, mathematical, chemical, or vital; production; formation; as, the generation of sounds, of gases, of curves, etc.
a.
Having the power of entering, piercing, or pervading; sharp; subtile; penetrative; as, a penetrating odor.
a.
Generating bile.
adv.
In all probability; probably.
pl.
of Probability
a.
Having the power of generating, propagating, originating, or producing.
n.
One who maintains that a man may do that which has a probability of being right, or which is inculcated by teachers of authority, although other opinions may seem to him still more probable.
n.
Likelihood of the occurrence of any event in the doctrine of chances, or the ratio of the number of favorable chances to the whole number of chances, favorable and unfavorable. See 1st Chance, n., 5.
n.
Probability.
n.
Probability; likelihood.
n.
Probability.
a.
Acute; discerning; sagacious; quick to discover; as, a penetrating mind.
n.
The doctrine of the probabilists.
n.
One who maintains that certainty is impossible, and that probability alone is to govern our faith and actions.
n.
The act of generating or begetting; procreation, as of animals.
n.
Probability; verisimilitude.
n.
Likelihood; probability.
superl.
Having probability; affording probability; probable; likely.
a.
Pertaining to generation, or to the generative organs.