An Introduction to Probability Theory and Its Applications, Volume II

Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II. Wiley.

Type

Book

Intellectual base

Textbook

Year

1971

Link

https://www.wiley.com/en-us/An+Introduction+to+Probability+Theory+and+Its+Applications%2C+Volume+2%2C+2nd+Edition-p-9780471257097

Cited from

mathematics, statistics_experimental_design

Cited by

6 citations across 6 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Central Limit Theorem
  • Replace the fair coin with a Cauchy-distributed contribution (infinite variance): the finite-variance precondition fails, and the sum no longer converges to a Gaussian — the average of \(n\) Cauchy draws is itself Cauchy, no narrower than a single draw, flowing to a stable-law attractor instead.
• Convolution
  • Signal processing, audio, and image. Low-pass and high-pass filters, Gaussian and median smoothing, edge detection via gradient kernels, and reverb via a room's impulse response — the entire filter literature is convolution with the right kernel. Probability. The density of \(X+Y\) for independent \(X, Y\) is the convolution of their densities; central-limit proofs are statements about iterated convolutions concentrating on the Gaussian, and characteristic-function methods are the Fourier-domain version.

    This sourceThe density of a sum of independent random variables is the convolution of their densities; characteristic functions multiply; iterated convolution concentrates on the Gaussian (central limit theorem).

• Inspection Paradox
  • The expected length of an interval seen by an arrival is the ratio of the second moment to the first, E[L²]/E[L], which equals or exceeds E[L], with equality only when all intervals are identical.

    This sourceDerives length-biased (size-biased) sampling and the waiting-time/inspection paradox, with the expected sampled interval equal to E[L^2]/E[L].

• Law of Large Numbers
  • The sample mean of n such draws has exactly the same distribution as a single draw, for every n, so averaging a million observations yields an estimate no more concentrated than one observation.

    This sourceEstablishes the stable-law property under which the mean of n Cauchy variates has the distribution of a single variate, so averaging does not concentrate.

• Renewal Process
  • That property is memorylessness, and the process it produces is the Poisson process: constant hazard, no ageing, and the only renewal process whose future is genuinely independent of how long the current interval has already run.

    This sourceChapter I.4 gives the lack-of-memory characterisation of the exponential and the waiting-time paradox; Chapter XI gives renewal theory under a finite mean, including the elementary renewal theorem.

Domain-specific

• Bates Distribution
  • The preserved Wikipedia revision is discovery provenance only; the identity is grounded in standard continuous-distribution and convolution theory.

Verification

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Links previously used in the corpus

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• https://www.wiley.com/en-us/An+Introduction+to+Probability+Theory+and+Its+Applications,+Volume+2,+2nd+Edition-p-9780471257097 ×1

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