An Introduction to Probability Theory and Its Applications, Volume II¶
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II. Wiley.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Central Limit Theorem
- 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).
- 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.
- 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].
- 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.
- 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.
- 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.
- 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.
- 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.
Domain-specific¶
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