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In probability theory, a probability space or a probability triple (,,) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: [1] [2]
In this case, the above formula applies, such as calculating the probability of a particular sum of the two rolls in an outcome. The probability of the event that the sum D 1 + D 2 {\displaystyle D_{1}+D_{2}} is five is 4 36 {\displaystyle {\frac {4}{36}}} , since four of the thirty-six equally likely pairs of outcomes sum to five.
Σ ∗ is a filtration of the underlying probability space (Ω, Σ, ); Y is adapted to the filtration Σ ∗, i.e., for each t in the index set T, the random variable Y t is a Σ t-measurable function; for each t, Y t lies in the L p space L 1 (Ω, Σ t, ; S), i.e.
That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset E {\displaystyle E\,} of the sample space Ω {\displaystyle \Omega \,} .
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
However, for a given sequence {X n} which converges in distribution to X 0 it is always possible to find a new probability space (Ω, F, P) and random variables {Y n, n = 0, 1, ...} defined on it such that Y n is equal in distribution to X n for each n ≥ 0, and Y n converges to Y 0 almost surely.
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.
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 ...