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We can calculate the probability P as the product of two probabilities: P = P 1 · P 2, where P 1 is the probability that the center of the needle falls close enough to a line for the needle to possibly cross it, and P 2 is the probability that the needle actually crosses the line, given that the center is within reach.
Let (,) be a metric space and consider two one-parameter families of probability measures on , say () > and () >. These two families are said to be exponentially equivalent if there exist a one-parameter family of probability spaces ( Ω , Σ ε , P ε ) ε > 0 {\displaystyle (\Omega ,\Sigma _{\varepsilon },P_{\varepsilon })_{\varepsilon >0}} ,
Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.
A random experiment is described or modeled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind. A mathematical description of an experiment consists of three parts: A sample space, Ω (or S), which is the set of all possible outcomes.
In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum.
Test whether A, B are statistically equivalent. If p is a real number such that 0 < p < 1, then produce a new ensemble by probabilistic sampling from A with probability p and from B with probability 1 − p. Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.
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