In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable...

formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set...

finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price...

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment...

μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often...

sub-probability) measures may not necessarily converge vaguely to a true probability measure, but rather to a sub-probability measure (a measure such...

theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always...

In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption...

be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume...

In probability theory, a probability space or a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is a mathematical construct...

(\Omega ,F,P)} is a probability space, with sample space Ω {\displaystyle \Omega } , event space F {\displaystyle F} and probability measure P {\displaystyle...

{\displaystyle \Sigma _{*}} and probability measure P {\displaystyle \mathbb {P} } if Σ∗ is a filtration of the underlying probability space (Ω, Σ,  P {\displaystyle...

{\displaystyle x\in S} ). The probability (with respect to some probability measure) that an event S {\displaystyle S} occurs is the probability that S {\displaystyle...

sense, this differs much from the modern meaning of probability, which in contrast is a measure of the weight of empirical evidence, and is arrived at...

measurable space ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} and probability measures P {\displaystyle P} and Q {\displaystyle Q} defined on ( Ω , F )...

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.) A random variable X {\displaystyle...

on the same probability space, the multivariate or joint probability distribution for X , Y , … {\displaystyle X,Y,\ldots } is a probability distribution...

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It...

with these properties are called σ-algebras. Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences...

for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space...

formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space (called the sample space)...

1A is the indicator function of A. The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome x in the...

{\displaystyle P} is a probability measure. p X ( x ) {\displaystyle p_{X}(x)} can also be simplified as p ( x ) {\displaystyle p(x)} . The probabilities associated...

heads or tails are given by the probabilities { p , 1 − p } {\displaystyle \{p,1-p\}} , then one can define a natural measure on the product space, given...

atoms and can have non-zero probabilities. Under the measure-theoretic definition of a probability space, the probability of an outcome need not even...

(the measure) is given by the probability. The total volume corresponds to probability one. This correspondence works because the axioms of probability theory...

In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome...

properties of such measures. However, when μ and ν are probability measures, the total variation distance of probability measures can be defined as ‖...

random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel. Consider two random...

In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability...