By P.E. Pfeiffer
It will be tricky to overestimate the significance of stochastic independence in either the theoretical improvement and the sensible appli cations of mathematical likelihood. the concept that is grounded within the concept that one occasion doesn't "condition" one other, within the feel that prevalence of 1 doesn't impact the possibility of the prevalence of the opposite. This ends up in a formula of the independence when it comes to an easy "product rule," that is amazingly winning in taking pictures the fundamental rules of independence. despite the fact that, there are numerous styles of "conditioning" encountered in perform which provide upward thrust to quasi independence stipulations. specific and distinct incorporation of those into the idea is required so as to take advantage of potent use of likelihood as a version for behavioral and actual structures. We research options of conditional independence. the 1st suggestion is kind of basic, using very trouble-free points of chance concept. basically algebraic operations are required to acquire fairly vital and priceless new effects, and to resolve many ambiguities and obscurities within the literature.
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Extra resources for Conditional Independence in Applied Probability
A, is to utilize the rule pee) > o. A, ~~ c. ,ith all the formal properties Bl-2 of the original, or prior, probability measure P(·). It sometimes happens that occurrence of the event the likelihood that assert that p(AIC) A will (or will not) occur. = peA) or p(AIC) C does not affect Thus, we may be able to = p(AIC c ). f one holds, so do the others. = peA) = peA) p(Aclc) = P (Ac ) P(AC\C c ) = P (Ac ) p(Alc) = p(Alc c ) = P(A)P(C) = P(A)P(C c ) peAce) = P(Ac)P(C) = P (C) = P(C c ) = P(C) P(CcIAc ) = P(C c ) P (AC) P(ACc) P (C \ A) P(CC\A) P(C\Ac ) p(AIC) P(A\C c ) p(Aclc) = p(Aclc c ) P(CIA) If any of these holds, we suppose the events pair, in a probabilistic sense.
Stochastic independence is a property of the probability assignment, hence is determined by the probability measure P(·). Change to a new probability measure may destroy the stochastic independence. ) The following extension of the contractor example shows how stochastic independence may fail to hold, even though the contractors work "independently" in an operational sense. It also leads to the concept of conditional independence. Example Bl-b Consider again the case of the two contractors. There ~ be some factor in the work situation which affects the performance of both.
6, and p(BcIQc) = P(Qc) = 1/2. 7. Initially, he Expert opinion assigns the p(WIQc)/p(wcIQc) of these assumptions, determine the odds p(AiQ) = 3/2. On the basis p(wl ABc)/p(wcl ABc) and the p(WIAB c ). probability B-lO A student is picked at random from a large freshman class in calculus. revious trigonometry course, A = event the student made grade "A" B = event the student made grade "B" on the first examination, or better in the course. 30. The student selected made "B" or better. p(TiB) b) P(A/T) What is the probability that the student had a previous course in'trigonometry?