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The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century. [4] [5] The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem. A probabilistic proof via n-dimensional Brownian motion is well known. [6] Non-probabilistic proofs were available earlier.
Khinchin's theorem (probability) Killing–Hopf theorem (Riemannian geometry) Kinoshita–Lee–Nauenberg theorem (quantum field theory) Kirby–Paris theorem (proof theory) Kirchberger's theorem (discrete geometry) Kirchhoff's theorem (graph theory) Kirszbraun theorem (Lipschitz continuity) Kleene fixed-point theorem (order theory)
It should only contain pages that are Probability theorems or lists of Probability theorems, as well as subcategories containing those things (themselves set categories). Topics about Probability theorems in general should be placed in relevant topic categories .
The Theorem data type can be easily implemented to optionally store proof objects, depending on the system's run-time configuration, so it generalizes the basic proof-generation approach. The design decision to use a general-purpose programming language for developing theorems means that, depending on the complexity of programs written, it is ...
The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics , mathematics , science and philosophy to draw conclusions about the likelihood of potential ...
In the latter two examples the law of total probability is irrelevant, since only a single event (the condition) is given. By contrast, in the example above the law of total probability applies, since the event X = 0.5 is included into a family of events X = x where x runs over (−1,1), and these events are a partition of the probability space.
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