Gerolamo Cardano

1
(No Transcript)
2
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
3
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
Pierre de Fermat
now regarded as joint founders of probability
theory.
France
17 August 1601 12 January 1665
Blaise Pasca
June 19, 1623 August 19, 1662
4
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.
5
Nicolaus Bernoulli (1623-1708)
Jakob Bernoulli (16541705)
Nicolaus Bernoulli (16621716)
Johann Bernoulli (16671748)
Nicolaus I Bernoulli (1687-1759)
Nicolaus II Bernoulli (16951726)
Daniel Bernoulli (17001782)
Johann II Bernoulli (17101790)
Johann III Bernoulli (17441807)
Daniel II Bernoulli (17511834)
Jakob II Bernoulli (17591789)
6
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.
7
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
8
The normal distribution, also called the Gaussian
distribution,
Johann Carl Friedrich Gauss
30 April 1777 23 February 1855
Germany
9
(No Transcript)
10
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.
11
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.
12
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.
13
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.
14
(No Transcript)
15
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
16
Buffon's Needle Problem
the French naturalist Buffon in 1733
17
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
p
by needle-tossing.
18
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(HE,I) p(HI)p(EH,I)/p(EI) Bayes
Rule
Thomas Bayes 1702 - 1761
p(HE,I), is called the posterior probability,
London, England
The p(HI) is just the prior probability of H
given I alone
19
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.
20
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.