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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.
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
A Bernoulli process is a finite or infinite sequence of independent random variables X 1, X 2, X 3, ..., such that . for each i, the value of X i is either 0 or 1;; for all values of , the probability p that X i = 1 is the same.
Let be a discrete random variable with probability mass function depending on a parameter .Then the function = = (=),considered as a function of , is the likelihood function, given the outcome of the random variable .
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.
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}} ,
The probability spaces of the product are invariant and the probability of a given sequence is the product of the probabilities at each trial. Consequently, if p=P(t) is the prior probability that the outcome is t and the number of experiments is ld we obtain the probability of X t = t f {\displaystyle X_{t}=tf} is equal to:
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.