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This is a good point, I mused about it too. First, Pythagoreans and Plato had a very high minded idea of mathematics, gambling would have been seen as a lowly pursuit. This in itself does not explain it however. Plato's successors at the Academy considered mechanics lowly too, for it "uses bodies needing much vulgar manual labour, mechanics was thus ...


6

It is interesting that even many of those who retell the anecdote immediately disown it "The following anecdote about him is probably fabricated, but it makes an interesting probability problem", says Actuarial Outpost. Oloffson who like Mlodinov even put it into his book (p.203), writes "Although the anecdote illustrates a clever way to use probabilities to ...


6

This question has no definitive answer, because people were operating with random variables long before any rigorous definition was given. Probability theory begins in 16-th century, if not earlier. Cardano wrote a book on it, for example. In 1773 de Moivre wrote an important book where he essentially introduced the principal method of modern probability (...


5

The general idea of generating function has much wider scope than its applications to probability. The proper setting is ``harmonic analysis'' which is one of the central and most developed parts of mathematics. The birth of the idea can be traced back to Abraham de Moivre (1667-1754), and his book Doctrine of Chances. Later the same idea was developed and ...


5

This is a continuous analog of (transposed) stochastic matrix, the transition matrix in a Markov chain with discrete set of outcomes. These were introduced in 1906 by Markov apparently to disprove Nekrasov's claim that central limit theorems only applied to independent events, but later found many practical applications. Entries of a stochastic matrix are ...


5

Concerning the notation $\text{Pr}(|\xi|>\varepsilon)$ here's what I've found so far: Cajori's 1929 A History of Mathematical Notations says nothing on probability theory, which suggest that the subject had not yet developed any special or widely adopted notation around the beginning of the 20th century. This seems to be supported by Jeff Miller, who ...


4

I assume that one of the sources is MathWorld. But the question they claim Avez attributes to Gelfand is not the distribution of the leading digits generally, but specifically "will the digit 9 ever occur" as the leading digit in $2^n$ (the answer is yes, but the smallest $n$ is 54). They also link to Avez's 1966 book, which is their source. I was unable to ...


4

Imho the Greeks would have considered probabilities as a sophism - the attempt to produce knowledge out of ignorance. And even today they are still not far from truth: except for the frequentist interpretation which is objective, the rest is subjective (mostly the Bayesian kind). Kolmogoroff axiomatic evades the interpretation problem but so it is also known ...


4

From Earliest use of mathematical symbols: The convergence in probability symbol plim was introduced by H. B. Mann and A. Wald "On Stochastic Limit and Order Relationships," Annals of Mathematical Statistics, 14, (1943), 217-226. The stochastic order symbols $O_p$ and $o_p$, modelled on the $O$ and $o$, or Landau symbols, were introduced in the same paper....


4

My answer taken from the closed version of this question here From MathWords Moment was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation curve) as the sheet enclosed by the curve and the horizontal axis. See his "Asymmetrical Frequency Curves," Nature October 26th 1893: "Now the centre of gravity of ...


3

Long before Pearson, Chebyshev and his student A. A. Markov used moments to prove the Central Limit Theorem. The earliest paper of Chebyshev on this topic is dated 1887. But I do not claim that Chebyshev "was the first". On my opinion, a question of the type "who was the first" is meaningless and usually impossible to answer. http://citeseerx.ist.psu.edu/...


3

The first uses of what we call "uniform distribution" occur very early, discrete arguably already in Cardano, and continuous in Simpson and Bayes. According to Handbook of Beta Distributions: "One of the first records that mentions the continuos uniform distribution is the famous paper by the reverend Thomas Bayes (1763) (only a few years after ...


3

See : Pierre Remond de Montmort (1713), Essay d'analyse sur les jeux de hazard, Extrait d'une Lettre de M.N.Bernoulli à M.de M... du 9 Septembre 1713 : Cinquiéme Problème. On demande la meme chose si A promet à B de lui donner des écus en cette progression $1, 2, 4, 8, 16$ etc. [Fifth Problem. One asks the same thing [see: Fourth Problem] if A promises ...


3

The idea of using Gaussian mixtures was popularized by Duda and Hart in their seminal 1973 text, Pattern Classification and Scene Analysis.


3

I hope another answer is okay. Absolutely nothing of what follows is mine. The source[1] is given at the end. That main concern of that article, however, is de Finetti's theory of subjective probability and Hume's problem of induction. It has been suggested that an important step for the development of ideas of probability is the notion of what Jerzy Neyman ...


3

After looking for a while this is what I have found, although it does not mean that perhaps earlier references do not exist. In the review article by M. Belloni and R. W. Robinett, "The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics", PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS ...


2

Perhaps the best known contributions are the method of a single probability space, and the Skorokhod integral, which extends the Itō integral to non-adapted processes and is adjoint to the Malliavin derivative. He also did some foundational work on stochastic processes on manifolds with boundary. The Institute of Mathematics in Kiev maintains a Skorokhod ...


2

The book of Avez you refer to is indeed rare: it is not listed in the common databases Mathscinet and Zentralblatt (which is very strange). So I cannot say anything about the relation of Gelfand to this problem. A better known book Arnold and Avez, Ergodic Problems of Classical Mechanics, has this problem as a problem 3.2 in Ch.1.3, with solution in ...


2

Regarding English, I think that the first treatise was Abraham De Moivre's treatise The Doctrine of Chances (1718). For the discrete case, see page 7 : "If the Events in question are $n$ in number and are such as have the same number $a$ of Chances by whcih they may Happen..." And see also William Emerson (1701 – 1782)'s The Laws of Chance (1776), ...


2

Bayesian inference seeks to believe in that which has a high conditional probability as computed with Bayes's theorem. The problem with using $P\left( A|B\right)=P\left( B|A\right)\frac{P\left( A\right)}{P\left( B\right)}$ to compute the LHS is that, although the first term on the RHS is often known, at least one of the RHS's other probabilities usually won'...


2

The answer is not so straightforward. Of course, the thrust of Popper's position was against probabilistic induction in general, and Bayesianism is often put forth as the leading alternative to falsificationism, see Hypothesis testing: Fisher vs. Popper vs. Bayes. In 1983 Popper (with Miller) even offered a technical argument published as a note A proof of ...


1

It looks like he is saying the probable error of the difference between the two proportions is no more than the sum of the two probable errors, so $$d_1+d_2=0.0008+0.0003=0.0011$$ and that $$10(d_1+d_2)=0.011\approx 0.0105=\text{the actual discrepancy.}$$


1

Please allow me to answer my own question by putting together information gleaned from wikipedia and here, I am reasonably confident that the following is correct. If you have stronger evidence, such as a local newspaper photo/sketch of that time period, please do post an answer and I'll accept it. The layout of the wheel in use at the Monte Carlo casino on ...


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If we have a trial with non-zero probability of success $p>0$, and we repeat the trial $n$ times, the probability of at least one success is $p(n)=1-(1-p)^n$, which converges to $1$ when $n\to\infty$. An example would be tossing a fair coin (with $p=1/2$), and expecting to get heads eventually. Even if the coin is loaded for tails as long as $p>0$ we ...


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Intriguingly, I have never used references prior to the eighties. The oldest I used were: R.G. Newton Inverse scattering by a local impurity in a periodic potential in one dimension, JMP 24 (1983) P. Seba, The generalized point interaction in one dimension, Cz J Phys B 36:667 (1986) M. Carreau; Four-parameter point-interaction in 1D-quantum systems, J ...


1

See Stigler history of Statistics, p. 92 in which the binomial comes from, Pascal, Newton, Bernoulli, and De Moivre in that order. Hth Best!


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