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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 (PMF), discrete probability density function, probability function, or frequency function is
Probability_mass_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
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
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
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
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
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
the probability generating function is, GY(z)=exp(λ(eϕ(z−1)−1)) From the generating function of probabilities we can calculate the probability mass
Neyman_Type_A_distribution
Discrete probability distribution
density functions of the binomial and Poisson distributions, one has: FB(k;n,p) ≈ FP(k;λ=np). One derivation of this uses probability-generating functions. Consider
Poisson_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
Probability distribution
The probability of getting exactly k successes in n independent Bernoulli trials (with the same rate p) is given by the probability mass function: f(k;n
Binomial_distribution
Maxwell's theorem Moment-generating function Factorial moment generating function Negative probability Probability-generating function Vysochanskiï–Petunin
List_of_probability_topics
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
Probability distribution
derivative: I(p)=−E[∂2∂p2lnL(p;X)]=−(−1p2−1−pp(1−p)2)=1p2(1−p) The probability generating functions of geometric random variables X and Y defined over ℕ and ℕ0
Geometric_distribution
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
Probability distribution
distribution for a real-valued random variable. The general form of its probability density function is f(x)=12πσ2exp(−(x−μ)22σ2). The parameter μ is the mean
Normal_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
Function related to statistics and probability theory
components of a vector. For a probability function (or probability density function) Pr[x | θ] that gives the probability (or probability density) of data x for
Likelihood_function
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. Provided that
Expected_value
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
Measure of variation in statistics
the expected value (the average) of a random variable X with probability density function f: μ ≡ E [ X ] = ∫ − ∞ + ∞ x f ( x ) d x . {\displaystyle \mu
Standard_deviation
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
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
Fourth standardized moment in statistics
refers to the degree of tailedness in the probability distribution of a real-valued, random variable in probability theory and statistics. Similar to skewness
Kurtosis
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
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
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
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
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
Compound probability distribution
characteristic function has the form φX(s)=Mπ(eis−1). Where Mπ is the moment generating function of the density. For the probability generating function, one obtains
Mixed_Poisson_distribution
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
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
Probability plot correlation coefficient plot Probability space Probability theory Probability-generating function Probable error Probit Probit model Procedural
List_of_statistics_articles
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
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
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
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
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
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
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
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
Average uncertainty in variable's states
the function log(1p(E)), where log is the logarithm, which gives 0 surprise when the probability of the event is 1. In fact, log is the only function that
Entropy_(information_theory)
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
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
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
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)
Family of continuous probability distributions
continuous probability distributions with support on (0,∞). Its probability density function is given by f(x;μ,λ)=λ2πx3exp(−λ(x−μ)22μ2x) for x>0, where
Inverse_Gaussian_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
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
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
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
Measure of the shape of a function
the second moment is the moment of inertia. In statistics, the function is a probability distribution, then the first moment is the expected value, the
Moment_(mathematics)
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 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
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
Machine learning framework
game generates μG, a probability distribution on the probability space Ω. This leads to the idea of a conditional GAN, where instead of generating one
Generative adversarial network
Generative_adversarial_network
Discrete probability distribution
T(x,ν)=∑k=0nxk(nk)ν. Then, the probability generating function, moment generating function and characteristic function are given, respectively, by: G(t)=T(tp/(1−p)
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
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 function having a characteristic S-shaped curve or sigmoid curve
distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The
Sigmoid_function
Artificial intelligence model developed by TypeSafe AI
not generation. It does not generate natural-language text. It instead returns typed values together with probability estimates and confidence scores
Jev_(AI_model)
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
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
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
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
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
Algorithm for shuffling a finite sequence
of the sequence with a much higher probability for than positions near the ends. Randomized comparison functions applied to other sorting methods like
Fisher–Yates_shuffle
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
Continuous function that is not absolutely continuous
represented as an integral of a probability density function; integrating any putative probability density function that is not almost everywhere zero
Cantor_function
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
Probabilistic optimization technique and metaheuristic
candidate moves are generated by the neighbour() function, and on the acceptance probability function P(). Note that the transition probability is not simply
Simulated_annealing
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
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)
Theory of behavioral economics
are framed. The function π is a probability weighting function and captures the idea that people tend to overreact to small probability events, but underreact
Prospect_theory
Probability distribution
over the variance parameter. Student's t distribution has the probability density function (PDF) given by f ( t ) = Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) ( 1 + t
Student's_t-distribution
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
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
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
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
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
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
Class of statistical models
exponential families of probability distributions, 2. A linear predictor η = X β {\displaystyle \eta =X\beta } , and 3. A link function g {\displaystyle g}
Generalized_linear_model
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
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
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
Power series theorem in mathematics
functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton–Watson
Abel's_theorem
Family of probability distributions
variable X {\displaystyle X} whose probability distribution function belongs to such a family, the distribution function of Y = d a + b X {\displaystyle
Location–scale_family
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
Aspect of probability theory
\left((\varphi _{X}(t))^{N}\right),\,} and hence, using the probability-generating function of the Poisson distribution, we have φ Y ( t ) = e λ ( φ X
Compound_Poisson_distribution
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
Term in statistical hypothesis testing
between the two distributions being a function of the effect size), the power of the test would be the probability, under H1, that the sample t falls into
Power_(statistics)
Measure of the shape of a probability distribution function
In statistics, the moments of a probability density function are measures related to the shape of the function's graph. The first moment is the expected
Moment_(statistics)
Probability distribution
{\displaystyle \operatorname {Laplace} (\mu ,b)} distribution if its probability density function is f ( x ∣ μ , b ) = 1 2 b e − | x − μ | b , {\displaystyle f(x\mid
Laplace_distribution
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
Probability distribution
distribution, then the probability that X is greater than some number x, i.e., the survival function (also called tail function), is given by F ¯ ( x )
Pareto_distribution
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PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
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PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
PROBABILITY GENERATING-FUNCTION
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