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The focus turns toward the detailed description of the law of total probability and Bayes’ theorem. Finally, some important applications in probability, including systems reliability, medical diagnostic testing, quality control using Monte Carlo algorithm, and Bayesian spam filtering, are discussed. ...
As a consequence of the proof of Theorem2.1and Corollary2.2, we can compute the probability of{\mathcal {P}}_{A ,B}under the double random current measure: Theorem 2.3 LetAandBbe contiguous sets of boundary vertices as in Corollary2.2, and letv_1, v_2, \dots , v_{2k}be a counterc...
Moreover, it is a fundamental concept in probability theory involving conditional probabilities. It is closely linked with Bayes Theorem, which is widely used in various fields. While calculators are available to compute probabilities using this law, its proof involves a thorough understanding of condi...
=⋃i(A∩Bi)=⋃i(A∩Bi)by the distributive law (Theorem 1.2). A∩BiA∩Bi BiBi Here is a typical scenario in which we use the law of total probability. We are interested in finding the probability of an eventAA, but we don't know how to findP(A)P(A)directly. Instead, we ...
The paper first points out the deficiency of the formula of total probability,regarding that it is inconvenient to use and tends to be misleading.And then,the paper proposes a modified formula together with the proof of its improvement.Application shows that the formula is easy to understand,easy...
The proof is elementary and composed of ingredients at the postgraduate level. Our result clarifies that when \\alpha =1 \\alpha =1 and X has a symmetric Pareto distribution, the optimal rate is n^{-1} n^{-1} rather than n^{-1} (\\ln n)^2 n^{-1} (\\ln n)^2 as ...
Finally, the proof of Theorem 4 is just a refinement of the proof of Theorem 2, using a distribution similar to the one in Case 4 of the proof of Theorem 3. 2. Preliminary results By Theorem 2, we only have to consider the case ⌈Δ+12⌉<m+23. Without loss of generality we ...
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. As a consequence of the preceding result, we obtain the positivity of the function G2 in (3.10) or (3.11) as a limiting case of Theorem 3.13; this is done by taking limits similar to what was done in (3.8). Finally, we remark that the results of Theorems 3.1 and 3.3 clearly ...