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In case (1) above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P(λ ∈ E | φ = 0). Elementary probability theory suggests this can be computed as P(λ ∈ E and φ = 0)/P(φ = 0), but that expression is not well-defined since P(φ = 0) = 0.
The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process.
The term measure here refers to the measure-theoretic approach to probability. Violations of unit measure have been reported in arguments about the outcomes of events [2] [3] under which events acquire "probabilities" that are not the probabilities of probability theory. In situations such as these the term "probability" serves as a false ...
In fact, the discrete case (although without the restriction to probability measures) is the first step in proving the general measure-theoretic formulation, as the general version follows therefrom by an application of the monotone convergence theorem. [7] Without any major changes, the result can also be formulated in the setting of outer ...
Many of the concepts in information theory have separate definitions and formulas for continuous and discrete cases. For example, entropy is usually defined for discrete random variables, whereas for continuous random variables the related concept of differential entropy, written (), is used (see Cover and Thomas, 2006, chapter 8).
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 additivity. [1] The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign ...
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality. [1]
When the underlying measure on (, ()) is finite, the distribution function in Definition 3 differs slightly from the standard definition of the distribution function (in the sense of probability theory) as given by Definition 2 in that for the former, = while for the latter, () = = ().
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