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In continuous distributions, the probability of any event... — Carmelics

Statements: 321,452
Perspectives: 108,905
Topics: 42

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In continuous distributions, the probability of any event can be computed via integration of a probability density function

Modality & Possibility

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1 reason for

2 reasons against

Reasons For

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Reason for

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1.In common continuous distributions there is usually a way to define a probability density for each state
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2.The probability of any event is the integral of the density over the states that make up the event
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Reasons Against

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Reason against 1 of 2

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1.Vitali sets and other non-measurable subsets of the real line cannot be assigned consistent probability values via Lebesgue integration.
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2.Any probability space over a continuous domain must exclude legitimate subsets, making 'any event' computability via integration demonstrably false under ZFC.
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Reason against 2 of 2

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1.De Finetti's constructivist approach shows that probability density functions presuppose a prior measure, making the claim circular for foundational purposes.
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2.Without an independently justified reference measure, the choice of density function is arbitrary, undermining the claim that probabilities are computed rather than stipulated.
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Modality & Possibility

Related

Any probability space over a continuous domain must exclude legitimate subsets, ...De Finetti's constructivist approach shows that probability density functions pr...In common continuous distributions there is usually a way to define a probabilit...The probability of any event is the integral of the density over the states that...

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Vitali sets and other non-measurable subsets of the real line cannot be assigned...Without an independently justified reference measure, the choice of density func...

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In common continuous distributions there is usually a way to define a ...84%In continuous distributions there is always some event of probability ...83%The probability of any event is the integral of the density over the s...80%The standard treatment of zero-probability events differs between fini...76%

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However, many applications of probability require what are known as continuous distributions (such as the uniform/rectangular, normal, and beta distributions), and thus require a restriction to countable additivity. In a continuous distribution, there are uncountably many states, usually named by real numbers. Each individual state has probability 0, even though events containing uncountably many states often have non-zero probability. (This violates full additivity.) However, in the common cont

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Validity: Extracted via Max plan + API grounding/validity checks

Details

Type: claim
Perspectives: 3 (1 for, 2 against)
Edits: 1 edit