Transcript 幻灯片 1

Cardano was notoriously short of money and kept
himself solvent by being an accomplished gambler
and chess player. His book about games of chance,
Liber de ludo aleae, written in the 1560s, but not
published until 1663, contains the first systematic
treatment of probability, as well as a section on
effective cheating methods.
Gerolamo Cardano
September 24, 1501 — September 21, 1576
was an Italian Renaissance mathematician, physician, astrologer and gambler
Through his correspondence with Blaise Pascal
in 1654, Fermat and Pascal helped lay the fundamental
groundwork for the theory of probability. From this brief but
productive collaboration on the problem of points, they are
now regarded as joint founders of probability theory.
Pierre de Fermat
France
17 August 1601 – 12 January 1665
Blaise Pasca
June 19, 1623 – August 19, 1662
Jacob is best known for the work
Ars Conjectandi (The Art of Conjecture),
published eight years after his death in 1713
In this work, he described the known results in
probability theory and in enumeration, often
providing alternative proofs of known results.
Jacob Bernoulli
This work also includes the application of
27 December 1654 – 16 August 1705
probability theory to games of chance and his
Basel, Switzerland
introduction of the theorem known as
the law of large numbers.
Nicolaus Bernoulli (1623-1708)
Jakob Bernoulli (1654–1705) Nicolaus Bernoulli (1662–1716)
Johann Bernoulli (1667–1748)
Nicolaus I Bernoulli (1687-1759)
Nicolaus II Bernoulli (1695–1726)
Daniel Bernoulli (1700–1782)
Johann II Bernoulli (1710–1790)
Johann III Bernoulli (1744–1807)
Daniel II Bernoulli (1751–1834)
Jakob II Bernoulli (1759–1789)
De Moivre wrote a book on probability theory,
entitled The Doctrine of Chances. It was said
that his book was highly prized by gamblers. It
is reported in all seriousness that de Moivre
correctly predicted the day of his own death.
Abraham de Moivre
Noting that he was sleeping 15 minutes longer
France
26 May 1667 – 27 November 1754
each day, De Moivre surmised that he would die
on the day he would sleep for 24 hours. A simple
mathematical calculation quickly yielded the
date, 27 November 1754. He did indeed pass away on that day.
In 1812, Laplace issued his
Théorie analytique des probabilités
in which he laid down many fundamental results in statistics.
Pierre-Simon Laplace
French
23 March 1749 - 5 March 1827
The normal distribution, also called the Gaussian distribution,
Johann Carl Friedrich Gauss
30 April 1777 – 23 February 1855
Germany
The normal distribution was first introduced by Abraham de Moivre
in an article in 1733, which was reprinted in the second edition of his
The Doctrine of Chances, 1738 in the context of approximating certain
binomial distributions for large n. His result was extended by Laplace
in his book Analytical Theory of Probabilities (1812), and is now called the
theorem of de Moivre-Laplace.
Laplace used the normal distribution in the analysis of errors of experiments.
Its usefulness, however, became truly apparent only in 1809, when the famous
German mathematician K.F. Gauss used it as an integral part of his
approach to prediction the location of astronomical entities. As a result,
it became common after this time to call it the Gaussian distribution.
During the mid to late nineteenth century, however, most statisticians started to
believe that the majority of data sets would have histograms conforming to the
Gaussian bell-shaped form. Indeed, it came to be accepted that it was “normal”
for any well-behaved data set to follow this curve. As a result, following
the lead of the British statistician Karl Pearson, people began referring to
the Gaussian curve to calling it simply the normal curve.
The name "bell curve" goes back to Esprit Jouffret who first used
the term "bell surface" in 1872 for a bivariate normal with independent
components. The name "normal distribution" was coined independently
by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875.
His monograph on probability theory
Grundbegriffe der Wahrscheinlichkeitsrechnung
published in 1933 built up probability theory in a rigorous way
from fundamental axioms in a way
comparable with Euclid's treatment of geometry.
Andrey Nikolaevich Kolmogorov
25 April 1903 -- 20 Oct 1987
Moscow, Russia
Buffon's Needle Problem
the French naturalist Buffon in 1733
Buffon's needle problem asks to find the probability that a needle of length
will land on a line, given a floor with equally spaced parallel lines a distance
apart. The problem was first posed by the French naturalist Buffon in 1733
(Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777
Several attempts have been made to experimentally determine π
by needle-tossing.
Bayes' theorem gives the rule for updating belief in a
Hypothesis H (i.e. the probability of H)
given additional evidence E, and background information (context) I
p(H|E,I) = p(H|I)*p(E|H,I)/p(E|I)
[Bayes Rule]
Thomas Bayes
p(H|E,I), is called the posterior probability,
1702 - 1761
London, England
The p(H|I) is just the prior probability of H given I alone
Bayes' theorem is particularly useful for inferring causes from their effects
since it is often fairly easy to discern the probability of an effect given the
presence or absence of a putative cause.
For instance, physicians often screen for diseases of known prevalence
using diagnostic tests of recognized sensitivity and specificity.
The sensitivity of a test, its "true positive" rate, is the fraction of times
that patients with the disease test positive for it.
The test's specificity, its "true negative" rate, is the proportion of healthy
patients who test negative.
one can use to determine the probability of disease given a positive test.
The essence of the Bayesian approach is to provide a mathematical rule
explaining how you should change your existing beliefs in the light of new
evidence. In other words, it allows scientists to combine new data with
their existing knowledge or expertise. The canonical example is to imagine that
a precocious newborn observes his first sunset, and wonders whether the sun will
rise again or not.