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Archive:Athena Parthenos

2008-10-03T02:54:32Z

71.113.133.196:

'''''Athena Parthenos''''' (Athena the Virgin) was the title of a massive [[chryselephantine]] [[sculpture]] of the [[Greek mythology|Greek]] [[goddess]] [[Athena]] by [[Phidias]]. It was named after an epithet for the goddess herself, and was housed in the [[Parthenon]] in [[Athens]]. A number of replicas and works inspired by it, both ancient and modern, have been made.

[[Image:NAMA Athéna Varvakeion.jpg|thumb|A votive sculpture found near the [[Ioannis Varvakis|Varvakeion school]] reflects the type of the restored ''Athena Parthenos'': Roman period, second century CE ([[National Archaeological Museum of Athens]]).]]
It was the most renowned [[cult image]] of Athens,<ref>The ''Athena Parthenos'' was featured on contemporary reliefs commemorating Athenian treaties and for the next century and a half on coins of [[Hellenistic]] monarchs avid to proclaim their Hellenic connections; see Hector Williams, "An Athena Parthenos from Cilicia" ''Anatolian Studies'' '''27''' (1977, pp. 105-110), p 108f.</ref> considered one of the greatest achievements of the most acclaimed sculptor of ancient Greece. Phidias began his work around 447 BC,<ref>Andrew Stewart gives 446.</ref> [[Lachares]] removed the gold sheets in 296 BC to pay his troops, and the bronze was probably gilded thereafter; it was damaged by a fire about 165 BC but repaired.<ref>W.B. Dinsmoor, "the repair of the Athena Parthenos" ''[[American Journal of Archaeology]]'' '''38''' (1934) pp 93-106.</ref> It continued to stand in the Parthenon in the fifth century AD, when it may have been lost in another fire. An account mentions it in Constantinople in the tenth century, however.<ref>[[Gisela Richter]], ''Sculpture and Sculptors of the Greeks'', p 220 with ancient references, noted by Gorham P. Stevens, "Concerning the Parthenos" ''Hesperia'' '''30'''.1 (January 1961, pp. 1-7) p. 2.</ref>

==Description==
The ancient historian [[Pausanias (geographer)|Pausanias]] gave a description of the statue:
<blockquote>...The statue itself is made of ivory silver and gold. On the middle of her helmet is placed a likeness of the [[Sphinx]] ... and on either side of the helmet are [[griffin]]s in relief. ... The statue of Athena is upright, with a tunic reaching to the feet, and on her breast the head of [[Medusa]] is worked in ivory. She holds a statue of [[Nike (mythology)|Victory]] about four [[cubit]]s high, and in the other hand a spear; at her feet lies a shield and near the spear is a serpent. This serpent would be [[Erichthonius of Athens|Erichthonius]]. On the pedestal is the birth of [[Pandora]] in relief.<ref>Pausanias, ''Description of Greece'', Book I, 24.5–7.</ref></blockquote>

Athena's head is inclined slightly forward. She stands with her left hand resting on an upright shield. Her left knee is slightly bent, her weight slightly shifted to her right leg. Her [[Chiton (costume)|chiton]] is cinched at the waist by a pair of [[Serpent (symbolism)|serpents]], whose tails entwine at the back. Locks of hair trail onto the goddess's breastplate. The [[Nike (mythology)|Nike]] on her outstretched right hand is winged; whether there was a support under it in Phidias' original has been much discussed;<ref>[[Gisela Richter]] decided there was not and summarized the discussion in "Was there a vertical support under the Nike of the Athena Parthenos?" ''Studi in onore... Calderini e Paribeni'' (Milan) 1956, pp 147-54.</ref> evidence in surviving versions is contradictory. The exact position of a spear, often omitted, is also not fully determined, whether held in the crook of Athena's right arm or supported by one of the snakes in the [[aegis]], as N. Leipen restores it,<ref>Leipen 1971:29.</ref> following the "Aspasios" gem.

The sculpture was assembled on a wooden core, covered with shaped bronze plates covered in turn with removable gold plates, save for the ivory surfaces of the goddess's face and arms; the gold weighed 44 [[Talent (measurement)|talent]]s, the equivalent of about {{convert|2500|lb|kg}}; the ''Athena Parthenos'' embodied a sizeable part of the treasury of Athens.<ref>S. Eddy, "The gold in the Athena Parthenos'' ''Merican Jopurnal of Archaeology'' pp107-11.</ref>

The general type of the ''Athena Parthenos'', although not its character and quality, can be assessed from its image on coins<ref>L. Lacroix ''Les reproductions des statues sur les monnaies gracques'' (Liège) 1949, pp 266-81.</ref> and its reproductions as [[#Ancient copies|miniature sculptures]], as [[Votive deposit|votive object]]s, and in representations on engraved gems.<ref>Leipen 1971.</ref>

==Ancient copies==
[[Image:Strangfrod shield pushkin.jpg|thumb|Plaster cast of the British Museum's Strangford Shield ([[Pushkin Museum]])]]
The Varvakeion votive figure (''illustration, above right'') is one of the two versions of ''Athena Parthenos'' considered most faithful to the original; the other is the uncompleted Lenormant Athena, also in the National Museum, Athens.
*The ''Varvakeion Athena'', a Roman copy in marble of the second century, is housed in the [[National Archaeological Museum of Athens]]. One of the two versions of ''Athena Parthenos'' considered most faithful to the original.<ref>http://74.125.95.104/u/ur?q=cache:4HJ8k7_FwW8J:oncampus.richmond.edu/academics/classics/students/Liu/parthenon.html+Athena+Parthenos&hl=en&ct=clnk&cd=1&ie=UTF-8</ref>
*Lenormant Athena, uncompleted, of the second to third century, also in the National Museum, Athens. One of the two versions of ''Athena Parthenos'' considered most faithful to the original.
*[[:Image:Athena Parthenos Louvre Ma91.jpg|Another copy]] is housed in the [[Louvre]].<ref>http://cartelen.louvre.fr/cartelen/visite?srv=car_not_frame&idNotice=809</ref>
*[[:Image:Athena Parthenos Altemps Inv8622.jpg|Another copy]] is in the [[Museo Nazionale Romano]] in Rome
*The statue's shield alone ([[Percy Smythe, 6th Viscount Strangford|Strangford Collection]], [[British Museum]], ''illustration, right'')<ref>http://www.britishmuseum.org/explore/highlights/highlight_objects/gr/f/fragment_of_a_marble_shield.aspx</ref>

== Replica at Nashville ==
[[Image:Nashville Parthenon 005.JPG|thumb|left|The modern ''Athena Parthenos'' replica that stands in the replicated [[Parthenon]] in [[Centennial Park (Nashville)|Centennial Park]], [[Nashville]]]]
A modern replica by [[Alan LeQuire]] stands in the [[Parthenon (Nashville)|reproduction of the Parthenon]] in [[Nashville]], [[Tennessee]]. Alan LeQuire, a Nashville native, was awarded the commission to produce the Parthenon's cult statue. His work was modeled on descriptions given of the original. The modern version took eight years to complete, and was unveiled to the public on [[May 20]] [[1990]].

The modern version of ''Athena Parthenos'' is significant because of its scale and its attention to recreating Phidias' work. The statue adds an additional dimension of realism to the replicated Parthenon, whose interior east room (the ''naos'') was merely a large empty hall prior to the statue's unveiling. The reproduced ''Athena Parthenos'' gives visitors the impression that they truly are inside an ancient place of worship.

''Athena Parthenos'' is made of a composite of [[gypsum]] cement and ground [[fiberglass]]. The head of Athena was assembled over an [[aluminum]] [[armature]], and the lower part was made in [[steel]]. The four ten-inch H beams rest on a concrete structure that extends through the Parthenon floor and basement down to [[bedrock]], to support the incredible weight of the statue. LeQuire made each of the 180 cast gypsum panels used to create the statue light enough to be lifted by one person and attached to the steel armature.

=== Gilding and paint ===
[[Image:AthenaAlanpainting.jpg|thumb|right|Sculptor [[Alan LeQuire]] painting the detail of the ''Athena Parthenos'' replica during the gilding phase.]]
Painstaking research was performed by LeQuire and the Parthenon staff to ensure the accuracy of the statue's resemblance to the Phidias work. It stood in Nashville’s Parthenon as a plain, white statue for twelve years. In [[2002]], Parthenon volunteers [[gild]]ed Athena under the supervision of master gilder Lou Reed. The gilding project took less than four months and makes the modern statue appear that much more like the Phidias' Athena Parthenos would have appeared during its time.

The gold plates on the Athena statue in ancient times weighed approximately {{convert|1500|lb|kg}} and were one-sixteenth to one-eighth of an inch (1.6 to 3.2 mm) thick. The 23.75-karat gold leaf on Nashville's Athena Parthenos weighs a total of {{convert|8.5|lb|kg}} and is one-third the thickness of tissue paper. The modern extravagance of gilding such a large statue pales in comparison to the lavish spending of the Greeks. In fact, one theory of the original's demise is that Athena Parthenos was decimated and looted to remove the gilding.

In addition to gilding, the project included painting the details of the statue's face, wardrobe, and shield. LeQuire himself applied the paint.

=== Facts and figures ===
* Nashville's Athena stands 41 ft 10 in ({{convert|502|in|m}}) tall, making her the largest piece of indoor sculpture in the Western World.
* The statue of Nike in Athena's right hand stands 6 ft 4 in ({{convert|76|in|m}}) tall.
* There are eleven snakes represented on Athena's breastplate, bracelets, and belt.

==Notes==

<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>

==Further reading==
{{commonscat|Athena Parthenos}}
*Leipen, N. ''Athena Parthenos'' (Toronto) 1971.

==External links==
*[http://www.perseus.tufts.edu/cl135/Students/Colin_Delaney/fathena.html Colin Delaney, "Pheidias, Sculptor to the Gods"]
* [http://alanlequire.com/athena.html Alan LeQuire Page]
* [http://www.nashville.gov/parthenon/index.htm Nashville Parthenon Page]

[[Category:Athena types]]
[[Category:Culture of Nashville, Tennessee]]
[[Category:1990 works]]
[[Category:Sculptures by Phidias]]
[[Category:Colossal statues]]
[[Category:Ivory works of art]]

[[ca:Atena Pàrtenos]]
[[es:Atenea Partenos]]
[[it:Atena Parthenos]]
[[ru:Афина Парфенос]]
[[zh:雅典娜·帕德嫩]]

Archive:Athena Parthenos

2008-10-03T02:50:44Z

71.113.133.196:

'''''Athena Parthenos''''' (Athena the Virgin) was the title of a massive [[chryselephantine]] [[sculpture]] of the [[Greek mythology|Greek]] [[goddess]] [[Athena]] by [[Phidias]]. It was named after an epithet for the goddess herself, and was housed in the [[Parthenon]] in [[Athens]]. A number of replicas and works inspired by it, both ancient and modern, have been made.

[[Image:NAMA Athéna Varvakeion.jpg|thumb|A votive sculpture found near the [[Ioannis Varvakis|Varvakeion school]] reflects the type of the restored ''Athena Parthenos'': Roman period, second century CE ([[National Archaeological Museum of Athens]]).]]
It was the most renowned [[cult image]] of Athens,<ref>The ''Athena Parthenos'' was featured on contemporary reliefs commemorating Athenian treaties and for the next century and a half on coins of [[Hellenistic]] monarchs avid to proclaim their Hellenic connections; see Hector Williams, "An Athena Parthenos from Cilicia" ''Anatolian Studies'' '''27''' (1977, pp. 105-110), p 108f.</ref> considered one of the greatest achievements of the most acclaimed sculptor of ancient Greece. Phidias began his work around 447 BC,<ref>Andrew Stewart gives 446.</ref> [[Lachares]] removed the gold sheets in 296 BC to pay his troops, and the bronze was probably gilded thereafter; it was damaged by a fire about 165 BC but repaired.<ref>W.B. Dinsmoor, "the repair of the Athena Parthenos" ''[[American Journal of Archaeology]]'' '''38''' (1934) pp 93-106.</ref> It continued to stand in the Parthenon in the fifth century CE, when it may have been lost in another fire. An account mentions it in Constantinople in the tenth century, however.<ref>[[Gisela Richter]], ''Sculpture and Sculptors of the Greeks'', p 220 with ancient references, noted by Gorham P. Stevens, "Concerning the Parthenos" ''Hesperia'' '''30'''.1 (January 1961, pp. 1-7) p. 2.</ref>

==Description==
The ancient historian [[Pausanias (geographer)|Pausanias]] gave a description of the statue:
<blockquote>...The statue itself is made of ivory silver and gold. On the middle of her helmet is placed a likeness of the [[Sphinx]] ... and on either side of the helmet are [[griffin]]s in relief. ... The statue of Athena is upright, with a tunic reaching to the feet, and on her breast the head of [[Medusa]] is worked in ivory. She holds a statue of [[Nike (mythology)|Victory]] about four [[cubit]]s high, and in the other hand a spear; at her feet lies a shield and near the spear is a serpent. This serpent would be [[Erichthonius of Athens|Erichthonius]]. On the pedestal is the birth of [[Pandora]] in relief.<ref>Pausanias, ''Description of Greece'', Book I, 24.5–7.</ref></blockquote>

Athena's head is inclined slightly forward. She stands with her left hand resting on an upright shield. Her left knee is slightly bent, her weight slightly shifted to her right leg. Her [[Chiton (costume)|chiton]] is cinched at the waist by a pair of [[Serpent (symbolism)|serpents]], whose tails entwine at the back. Locks of hair trail onto the goddess's breastplate. The [[Nike (mythology)|Nike]] on her outstretched right hand is winged; whether there was a support under it in Phidias' original has been much discussed;<ref>[[Gisela Richter]] decided there was not and summarized the discussion in "Was there a vertical support under the Nike of the Athena Parthenos?" ''Studi in onore... Calderini e Paribeni'' (Milan) 1956, pp 147-54.</ref> evidence in surviving versions is contradictory. The exact position of a spear, often omitted, is also not fully determined, whether held in the crook of Athena's right arm or supported by one of the snakes in the [[aegis]], as N. Leipen restores it,<ref>Leipen 1971:29.</ref> following the "Aspasios" gem.

The sculpture was assembled on a wooden core, covered with shaped bronze plates covered in turn with removable gold plates, save for the ivory surfaces of the goddess's face and arms; the gold weighed 44 [[Talent (measurement)|talent]]s, the equivalent of about {{convert|2500|lb|kg}}; the ''Athena Parthenos'' embodied a sizeable part of the treasury of Athens.<ref>S. Eddy, "The gold in the Athena Parthenos'' ''Merican Jopurnal of Archaeology'' pp107-11.</ref>

The general type of the ''Athena Parthenos'', although not its character and quality, can be assessed from its image on coins<ref>L. Lacroix ''Les reproductions des statues sur les monnaies gracques'' (Liège) 1949, pp 266-81.</ref> and its reproductions as [[#Ancient copies|miniature sculptures]], as [[Votive deposit|votive object]]s, and in representations on engraved gems.<ref>Leipen 1971.</ref>

==Ancient copies==
[[Image:Strangfrod shield pushkin.jpg|thumb|Plaster cast of the British Museum's Strangford Shield ([[Pushkin Museum]])]]
The Varvakeion votive figure (''illustration, above right'') is one of the two versions of ''Athena Parthenos'' considered most faithful to the original; the other is the uncompleted Lenormant Athena, also in the National Museum, Athens.
*The ''Varvakeion Athena'', a Roman copy in marble of the second century, is housed in the [[National Archaeological Museum of Athens]]. One of the two versions of ''Athena Parthenos'' considered most faithful to the original.<ref>http://74.125.95.104/u/ur?q=cache:4HJ8k7_FwW8J:oncampus.richmond.edu/academics/classics/students/Liu/parthenon.html+Athena+Parthenos&hl=en&ct=clnk&cd=1&ie=UTF-8</ref>
*Lenormant Athena, uncompleted, of the second to third century, also in the National Museum, Athens. One of the two versions of ''Athena Parthenos'' considered most faithful to the original.
*[[:Image:Athena Parthenos Louvre Ma91.jpg|Another copy]] is housed in the [[Louvre]].<ref>http://cartelen.louvre.fr/cartelen/visite?srv=car_not_frame&idNotice=809</ref>
*[[:Image:Athena Parthenos Altemps Inv8622.jpg|Another copy]] is in the [[Museo Nazionale Romano]] in Rome
*The statue's shield alone ([[Percy Smythe, 6th Viscount Strangford|Strangford Collection]], [[British Museum]], ''illustration, right'')<ref>http://www.britishmuseum.org/explore/highlights/highlight_objects/gr/f/fragment_of_a_marble_shield.aspx</ref>

== Replica at Nashville ==
[[Image:Nashville Parthenon 005.JPG|thumb|left|The modern ''Athena Parthenos'' replica that stands in the replicated [[Parthenon]] in [[Centennial Park (Nashville)|Centennial Park]], [[Nashville]]]]
A modern replica by [[Alan LeQuire]] stands in the [[Parthenon (Nashville)|reproduction of the Parthenon]] in [[Nashville]], [[Tennessee]]. Alan LeQuire, a Nashville native, was awarded the commission to produce the Parthenon's cult statue. His work was modeled on descriptions given of the original. The modern version took eight years to complete, and was unveiled to the public on [[May 20]] [[1990]].

The modern version of ''Athena Parthenos'' is significant because of its scale and its attention to recreating Phidias' work. The statue adds an additional dimension of realism to the replicated Parthenon, whose interior east room (the ''naos'') was merely a large empty hall prior to the statue's unveiling. The reproduced ''Athena Parthenos'' gives visitors the impression that they truly are inside an ancient place of worship.

''Athena Parthenos'' is made of a composite of [[gypsum]] cement and ground [[fiberglass]]. The head of Athena was assembled over an [[aluminum]] [[armature]], and the lower part was made in [[steel]]. The four ten-inch H beams rest on a concrete structure that extends through the Parthenon floor and basement down to [[bedrock]], to support the incredible weight of the statue. LeQuire made each of the 180 cast gypsum panels used to create the statue light enough to be lifted by one person and attached to the steel armature.

=== Gilding and paint ===
[[Image:AthenaAlanpainting.jpg|thumb|right|Sculptor [[Alan LeQuire]] painting the detail of the ''Athena Parthenos'' replica during the gilding phase.]]
Painstaking research was performed by LeQuire and the Parthenon staff to ensure the accuracy of the statue's resemblance to the Phidias work. It stood in Nashville’s Parthenon as a plain, white statue for twelve years. In [[2002]], Parthenon volunteers [[gild]]ed Athena under the supervision of master gilder Lou Reed. The gilding project took less than four months and makes the modern statue appear that much more like the Phidias' Athena Parthenos would have appeared during its time.

The gold plates on the Athena statue in ancient times weighed approximately {{convert|1500|lb|kg}} and were one-sixteenth to one-eighth of an inch (1.6 to 3.2 mm) thick. The 23.75-karat gold leaf on Nashville's Athena Parthenos weighs a total of {{convert|8.5|lb|kg}} and is one-third the thickness of tissue paper. The modern extravagance of gilding such a large statue pales in comparison to the lavish spending of the Greeks. In fact, one theory of the original's demise is that Athena Parthenos was decimated and looted to remove the gilding.

In addition to gilding, the project included painting the details of the statue's face, wardrobe, and shield. LeQuire himself applied the paint.

=== Facts and figures ===
* Nashville's Athena stands 41 ft 10 in ({{convert|502|in|m}}) tall, making her the largest piece of indoor sculpture in the Western World.
* The statue of Nike in Athena's right hand stands 6 ft 4 in ({{convert|76|in|m}}) tall.
* There are eleven snakes represented on Athena's breastplate, bracelets, and belt.

==Notes==

<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>

==Further reading==
{{commonscat|Athena Parthenos}}
*Leipen, N. ''Athena Parthenos'' (Toronto) 1971.

==External links==
*[http://www.perseus.tufts.edu/cl135/Students/Colin_Delaney/fathena.html Colin Delaney, "Pheidias, Sculptor to the Gods"]
* [http://alanlequire.com/athena.html Alan LeQuire Page]
* [http://www.nashville.gov/parthenon/index.htm Nashville Parthenon Page]

[[Category:Athena types]]
[[Category:Culture of Nashville, Tennessee]]
[[Category:1990 works]]
[[Category:Sculptures by Phidias]]
[[Category:Colossal statues]]
[[Category:Ivory works of art]]

[[ca:Atena Pàrtenos]]
[[es:Atenea Partenos]]
[[it:Atena Parthenos]]
[[ru:Афина Парфенос]]
[[zh:雅典娜·帕德嫩]]

Introduction to Statistics

2008-07-05T20:47:51Z

71.113.250.54: /* Examples */

{{nav|School:Mathematics/Undergraduate/Probability and Statistics}}

==Examples==
*46% of people polled enjoy vanilla, while 54% prefer chocolate (+/-4% margin of error).
*A school's graduation rate has increased by 2%.
*A couple has 4 boys, and they are pregnant again: what is their chance of having another boy?
*88% of people questioned feel that it is humane to put stray animals to sleep.

These are basic examples of statistics we see everyday, but do we really understand what they mean? With the study of statistics, these 'facts' that we hear everyday can hopefully become a little more clear.

== Preamble ==
Statistics is permeated by probability. An understanding of basic probability is critical for the understanding of the basic mathematical underpinning of statistics. Strictly speaking the word 'statistics' means one or more measures describing the characteristics of a population. We use the term here in a more idiomatic sense to mean everything to do with sampling and the establishment of population measures.

Most statistical procedures use probability to make a statement about the relationship between the independent variables and the dependent variables. Typically, the question one attempts to answer using statistics is that there is a relationship between two variables. To demonstrate that there is a relationship the experimenter must show that when one variable changes the second variable changes and that the amount of change is more than would be likely from mere chance alone.

There are two ways to figure the probability of an event. The first is to do a mathematical calculation to determine how often the event can happen. The second is to observe how often the event happens by counting the number of times the event could happen and also counting the number of times the event actually does happen.

The use of a mathematical calculation is when a person can say that the chance of the event rolling a one on a six sided die is one in six. The probability is figured by figuring the number of ways the event can happen and divide that number by the total number of possible outcomes. Another example is in a well shuffled deck of cards, what is the probability of the event of drawing a three. The answer is four in fifty two since there are four cards numbered three and there are a total of fifty two cards in a deck. The chance of the event of drawing a card in the suite of diamonds is thirteen in fifty two (there are thirteen cards of each of the four suites). The chance the event of drawing the three of diamonds is one in fifty two.

Sometimes, the size of the total event space, the number of different possible events, is not known. In that case, you will need to observe the event system and count the number of times the event actually happens versus the number of times it could happen but doesn't.

For instance, a warranty for a coffee maker is a probability statement. The manufacturer calculates that the probability the coffee maker will stop working before the warranty period ends is low. The way such a warranty is calculated involves testing the coffee maker to calculate how long the typical coffee maker continues to function. Then the manufacturer uses this calculation to specify a warranty period for the device. The actual calculation of the coffee maker's life span is made by testing coffee makers and the parts that make up a coffee maker and then using probability to calculate the warranty period.

== Experiments, Outcomes and Events ==

The easiest way to think of probability is in terms of experiments and their potential outcomes. Many examples can be drawn from everyday experience: On the drive home from work, you can encounter a flat tire, or have an uneventful drive; the outcome of an election can include either a win by candidate A, B, or C, or a runoff.

'''Definition:''' The entire collection of possible outcomes from an experiment is termed the ''sample space'', indicated as '''<math>\Omega</math>''' (''Omega'')

The simplest (albeit uninteresting) example would be an experiment with only one possible outcome, say <math>A</math>. From elementary set theory, we can express the sample space as follows:

<math>\Omega = \{ A \} </math>

A more interesting example is the result of rolling a six sided dice. The sample space for this experiment is:

<math>\Omega = \{ 1,2,3,4,5,6 \}</math>

We may be interested in ''events'' in an experiment.

'''Definition:''' An ''event'' is some subset of outcomes from the ''sample space''

In the dice example, events of interest might include 
a) the outcome is an even number 
b) the outcome is less than three 

These events can be expressed in terms of the possible outcomes from the experiment: 
a) : <math> \{2,4,6\} </math> 
b) : <math> \{ 1,2 \}</math>

We can borrow definitions from set theory to express events in terms of outcomes. Here is a refresher of some terminology, and some new terms that will be important later: 
<math> \cup </math> represents the Union of two events 
[[Image:Venn_A_union_B.png]] 
<math> \cap </math> represents the Intersection of two events 
[[Image:Venn_A_intersect_B.svg]] 
<math>\{\cdots\}^{c}</math> represents the complement of an event. For instance, "the outcome is an even number" is the complement of "the outcome is an odd number" in the dice example. 
<math> A \backslash B </math> represents ''difference'', that is, <math>A</math> ''but not'' <math>B</math>. For example, we may be interested in the event of drawing the queen of spades from a deck of cards. This can be expressed as the event of drawing a queen, but not drawing a queen of hearts, diamonds or clubs. 
<math>\varnothing</math> or <math>\{\}</math> represent an ''impossible event'' 
<math>\Omega</math> represents a ''certain event'' 
<math>A</math> and <math>B</math> are called ''disjoint events'' if <math>A\cap B = \varnothing</math>

== Probability ==
Now that we know what events are, we should think a bit about a way to express the likelihood of an event occurring. The classical definition of probability comes from the following. If we can perform our experiment over and over in a way that is repeatable, we can count the number of times that the experiment gives rise to event <math>A</math>. We also keep track of the number of times that we perform the same experiment. If we repeat the experiment a large enough number of times, we can express the probability of event <math>A</math> as follows: 
<math>P(A) = \frac{N_{A}}{N} </math> 
where <math>N_{A}</math> is the number of times event <math>A</math> occurred, and <math>N</math> is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event <math>A</math> equals the number of times event <math>A</math> occurs divided by the number of times the experiment was repeated (or the number of times event <math>A</math> ''could have'' occurred)." As <math>N</math> approaches infinity, the fraction above approaches the true probability of the event <math>A</math>. The value of <math>P(A)</math> is clearly between 0 and 1. If our event is the ''certain event'' <math>\Omega</math>, then for each time we perform the experiment, the event <math>\Omega</math> is observed; <math>N_{\Omega} = N</math> and <math>P(\Omega)=1</math>. If our event is the ''impossible event'' <math>\varnothing</math>, we know <math>N_{\varnothing}=0</math> and <math> P(\varnothing) = 0</math>.

If <math>A</math> and <math>B</math> are ''disjoint events'', then whenever event <math>A</math> is observed, then it is impossible for event <math>B</math> to be observed simultaneously. Therefore the number of times events <math>A</math> union <math>B</math> occurs are equal to the number of times event <math>A</math> occurred plus the number of times <math>B</math> occurs. This can be expressed as: 
<math>N(A\cup B) = N(A) + N(B)</math> 
Given our definition of probability, we can arrive at the following: 
<math>P(A\cup B) = P(A) + P(B)</math> 

At this point it's worth remembering that not all events are disjoint events. For events that are not disjoint, we end up with the following probability definition. 
<math> P(A\cup B) = P(A) + P(B) - P(A\cap B)</math> 
How can we see this from example? Well, let's consider drawing from a deck of cards. I'll define two ''events'': "drawing a Queen", and "drawing a Spade". It is immediately clear that these are not disjoint events, because you can draw a queen that is also a spade. There are four queens in the deck, so if we perform the experiment of drawing a card, putting it back in the deck and shuffling (what statisticians refer to as ''sampling with replacement'', we will end up with a probability of <math>\frac{1}{13}</math> for a queen draw. By the same argument, we obtain a probability for drawing a spade as <math>\frac{1}{4}</math>. The expression <math>P(A\cup B)</math> here can be translated as "the chance of drawing a queen or a spade". If we incorrectly assume that for this case <math>P(A\cup B) = P(A) + P(B)</math>, we can simply add our probabilities together for "the chance of drawing a queen or a spade" as <math>\frac{1}{13}+\frac{1}{4}</math>. If we were to gather some data experimentally, we would find that our results would differ from the prediction -- the probability observed would be slightly less than <math>\frac{1}{13}+\frac{1}{4}</math>. Why? Because we're counting the queen of spades twice in our expression, once as a spade, and again as a queen. We need to count it only once, as it can only be drawn with probability of <math>\frac{1}{52}</math>. [[Introduction to Statistics/Still confused?|Still confused?]] 

'''Proof:''' If <math>A</math> and <math>B</math> are not disjoint, we have to avoid the double counting problem by exactly specifying their union. 
<math>A \cup B = A \cup (B \backslash A) </math> so 
<math>P(A \cup B) = P(A \cup (B \backslash A)) </math> 
<math>A</math> and <math>B \backslash A</math> are disjoint sets. We can then use the definition of disjoint events from above to express our desired result: 
<math>P(A \cup B) = P(A) + P(B \backslash A)</math> 
We also know that 
<math>P(B \backslash A) = P(B) - P(B\cap A)</math> 
so 
<math> P(A \cup B) = P(A) + P(B) - P(B\cap A)</math> 
Whew! Our first proof. I hope that wasn't too [[dry]].

== Conditional Probability ==
Many events are conditional on the occurance of other events. Sometimes this coupling is weak. One event may become more or less probable depending on our knowledge that another event has occured. For instance, the probability that your friends and relatives will call asking for money is likely to be higher if you win the lottery. In my case, I don't think this probability would change.

Let's get formal for a second and remember our original definition of probability. 
<math>P(A) = \frac{N_{A}}{N}</math>
Consider an additional event <math>B</math>, and a situation where we are only interested in the probability of the occurrence of <math>A</math> when <math>B</math> occurs. A way at this probability is to perform a set of experiments (''trials'') and only record our results when the event <math>B</math> occurs. In other words 
<math> \frac{N_{A\cap B}}{N_{B}} </math> 
We can divide through on top and bottom by <math>N</math> the total number of trials to get <math>P(A\cap B)/P(B)</math>. We define this as ''''conditional probability'''': 
<math>P(A|B) = \frac{P(A\cap B)}{P(B)}</math> 
which when spoken, takes the sound "probability of <math>A</math> given <math>B</math>."

[[/A confusing problem/]]

=== Bayes' Law ===
'''THIS THEOREM IS NOT GOOD, THERE IS AN ERROR !!!!!!!''' 
'''JUST SEE THE RIGHT ONE AT THIS PAGE:''' 
http://en.wikipedia.org/wiki/Bayes'_theorem 
An important theorem in statistics is '''Bayes' Law''', which states that 
<math> P(B|A) = \frac{P(A|B)P(A)}{P(B)}</math>, 
It is easy to prove. We start with identical expressions for <math>P(A\cap B)</math>. 
We know that: <math> P(A\cap B) = P(B\cap A) </math>, 
<math> P(A\cap B) = \frac{P(A|B)}{P(B)}</math>, and 
<math> P(B\cap A) = \frac{P(B|A)}{P(A)}</math>. 
Since <math> P(A\cap B) = P(B\cap A) </math>, 
<math>\frac{P(A|B)}{P(B)} = \frac{P(B|A)}{P(A)}</math>. 
A Simple rearrangement of above line gives us '''Bayes' Law'''.

== Independence ==

Two events <math>A</math> and <math>B</math> are called ''independent'' if the occurence of one has absolutely no effect on the probability of the occurence of the other. Mathematically, this is expressed as: 
<math>P(A\cap B) = P(A)P(B)</math>. 

== Random Variables ==

It's usually possible to represent the outcome of experiments in terms of integers or real numbers. For instance, in the case of conducting a poll, it becomes a little cumbersome to present the outcomes of each individual respondant. Let's say we poll ten people for their voting preferences (Republican - R, or Democrat - D) in two different electorial districts. Our results might look like this: 
<math>\{RRRDRRDRRR\}</math> and <math>\{DDDDDRDDDD\}</math> 
But we're probably only interested in the overall breakdown in voting preference for each district. If we assign an integer value to each outcome, say 0 for Democrat and 1 for Republican, we can obtain a concise summary of voting preference by district simply by adding the results together.

== Discrete and Continuous Random Variables ==
There are two important subclasses of random variables: discrete random variable (DRV) and continuous random variable (CRV).
Discrete random variables take only countably many values. It means that we can list the set of all possible values that a discrete random variable can take, or in other words, the number of possible values in the set that the variable can take is finite. If the possible values that a DRV X can take are a0,a1,a2,...an, the probability that X takes each is p0=P(X=a0), p1=P(X=a1), p2=P(X=a2),...pn=P(X=an). All these probabilites are greater than or equal zero.

For continuous random variables, we cannot list all possible values that a continuous variable can take because the number of values it can take is extremely large. It means that there is no use to calculate the probability of each value separately because the probability that the variable takes a particular value is extremely small and can be considered zero P(X=x)=0).

== Distribution Functions ==

== Expectation Values ==

==See also==
* [[Topic:Statistics]]
* [[Introduction to research]]
* [[Statistical Economics]]
* [[Topic:Actuarial mathematics]]
* [[Introduction to Likelihood Theory]]
* [[Introduction to Classical Statistics]]
* [[Wikiversity:Statistics 202]]
* [[Introduction to probability and statistics]]
* [[Bayesian Statistics]]
* [[Statistics for Business Decisions]]
* [[Wikiversity:Statistics]]

[[Category:Mathematics]]
[[Category:Statistics]]
[[Category:Introductions]]

[[fr:Initiation à la statistique]]

Introduction to Statistics

2008-07-05T20:47:17Z

71.113.250.54: /* Examples */

{{nav|School:Mathematics/Undergraduate/Probability and Statistics}}

==Examples==
*46% of people polled enjoy vanilla, while 54% prefer chocolate (+/-4% margin of error).
*A school's graduation rate has increased by 2%.
*A couple has 4 boys, and they are pregnant again: what is the chance of having another boy?
*88% of people questioned feel that it is humane to put stray animals to sleep.

These are basic examples of statistics we see everyday, but do we really understand what they mean? With the study of statistics, these 'facts' that we hear everyday can hopefully become a little more clear.

== Preamble ==
Statistics is permeated by probability. An understanding of basic probability is critical for the understanding of the basic mathematical underpinning of statistics. Strictly speaking the word 'statistics' means one or more measures describing the characteristics of a population. We use the term here in a more idiomatic sense to mean everything to do with sampling and the establishment of population measures.

Most statistical procedures use probability to make a statement about the relationship between the independent variables and the dependent variables. Typically, the question one attempts to answer using statistics is that there is a relationship between two variables. To demonstrate that there is a relationship the experimenter must show that when one variable changes the second variable changes and that the amount of change is more than would be likely from mere chance alone.

There are two ways to figure the probability of an event. The first is to do a mathematical calculation to determine how often the event can happen. The second is to observe how often the event happens by counting the number of times the event could happen and also counting the number of times the event actually does happen.

The use of a mathematical calculation is when a person can say that the chance of the event rolling a one on a six sided die is one in six. The probability is figured by figuring the number of ways the event can happen and divide that number by the total number of possible outcomes. Another example is in a well shuffled deck of cards, what is the probability of the event of drawing a three. The answer is four in fifty two since there are four cards numbered three and there are a total of fifty two cards in a deck. The chance of the event of drawing a card in the suite of diamonds is thirteen in fifty two (there are thirteen cards of each of the four suites). The chance the event of drawing the three of diamonds is one in fifty two.

Sometimes, the size of the total event space, the number of different possible events, is not known. In that case, you will need to observe the event system and count the number of times the event actually happens versus the number of times it could happen but doesn't.

For instance, a warranty for a coffee maker is a probability statement. The manufacturer calculates that the probability the coffee maker will stop working before the warranty period ends is low. The way such a warranty is calculated involves testing the coffee maker to calculate how long the typical coffee maker continues to function. Then the manufacturer uses this calculation to specify a warranty period for the device. The actual calculation of the coffee maker's life span is made by testing coffee makers and the parts that make up a coffee maker and then using probability to calculate the warranty period.

== Experiments, Outcomes and Events ==

The easiest way to think of probability is in terms of experiments and their potential outcomes. Many examples can be drawn from everyday experience: On the drive home from work, you can encounter a flat tire, or have an uneventful drive; the outcome of an election can include either a win by candidate A, B, or C, or a runoff.

'''Definition:''' The entire collection of possible outcomes from an experiment is termed the ''sample space'', indicated as '''<math>\Omega</math>''' (''Omega'')

The simplest (albeit uninteresting) example would be an experiment with only one possible outcome, say <math>A</math>. From elementary set theory, we can express the sample space as follows:

<math>\Omega = \{ A \} </math>

A more interesting example is the result of rolling a six sided dice. The sample space for this experiment is:

<math>\Omega = \{ 1,2,3,4,5,6 \}</math>

We may be interested in ''events'' in an experiment.

'''Definition:''' An ''event'' is some subset of outcomes from the ''sample space''

In the dice example, events of interest might include 
a) the outcome is an even number 
b) the outcome is less than three 

These events can be expressed in terms of the possible outcomes from the experiment: 
a) : <math> \{2,4,6\} </math> 
b) : <math> \{ 1,2 \}</math>

We can borrow definitions from set theory to express events in terms of outcomes. Here is a refresher of some terminology, and some new terms that will be important later: 
<math> \cup </math> represents the Union of two events 
[[Image:Venn_A_union_B.png]] 
<math> \cap </math> represents the Intersection of two events 
[[Image:Venn_A_intersect_B.svg]] 
<math>\{\cdots\}^{c}</math> represents the complement of an event. For instance, "the outcome is an even number" is the complement of "the outcome is an odd number" in the dice example. 
<math> A \backslash B </math> represents ''difference'', that is, <math>A</math> ''but not'' <math>B</math>. For example, we may be interested in the event of drawing the queen of spades from a deck of cards. This can be expressed as the event of drawing a queen, but not drawing a queen of hearts, diamonds or clubs. 
<math>\varnothing</math> or <math>\{\}</math> represent an ''impossible event'' 
<math>\Omega</math> represents a ''certain event'' 
<math>A</math> and <math>B</math> are called ''disjoint events'' if <math>A\cap B = \varnothing</math>

== Probability ==
Now that we know what events are, we should think a bit about a way to express the likelihood of an event occurring. The classical definition of probability comes from the following. If we can perform our experiment over and over in a way that is repeatable, we can count the number of times that the experiment gives rise to event <math>A</math>. We also keep track of the number of times that we perform the same experiment. If we repeat the experiment a large enough number of times, we can express the probability of event <math>A</math> as follows: 
<math>P(A) = \frac{N_{A}}{N} </math> 
where <math>N_{A}</math> is the number of times event <math>A</math> occurred, and <math>N</math> is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event <math>A</math> equals the number of times event <math>A</math> occurs divided by the number of times the experiment was repeated (or the number of times event <math>A</math> ''could have'' occurred)." As <math>N</math> approaches infinity, the fraction above approaches the true probability of the event <math>A</math>. The value of <math>P(A)</math> is clearly between 0 and 1. If our event is the ''certain event'' <math>\Omega</math>, then for each time we perform the experiment, the event <math>\Omega</math> is observed; <math>N_{\Omega} = N</math> and <math>P(\Omega)=1</math>. If our event is the ''impossible event'' <math>\varnothing</math>, we know <math>N_{\varnothing}=0</math> and <math> P(\varnothing) = 0</math>.

If <math>A</math> and <math>B</math> are ''disjoint events'', then whenever event <math>A</math> is observed, then it is impossible for event <math>B</math> to be observed simultaneously. Therefore the number of times events <math>A</math> union <math>B</math> occurs are equal to the number of times event <math>A</math> occurred plus the number of times <math>B</math> occurs. This can be expressed as: 
<math>N(A\cup B) = N(A) + N(B)</math> 
Given our definition of probability, we can arrive at the following: 
<math>P(A\cup B) = P(A) + P(B)</math> 

At this point it's worth remembering that not all events are disjoint events. For events that are not disjoint, we end up with the following probability definition. 
<math> P(A\cup B) = P(A) + P(B) - P(A\cap B)</math> 
How can we see this from example? Well, let's consider drawing from a deck of cards. I'll define two ''events'': "drawing a Queen", and "drawing a Spade". It is immediately clear that these are not disjoint events, because you can draw a queen that is also a spade. There are four queens in the deck, so if we perform the experiment of drawing a card, putting it back in the deck and shuffling (what statisticians refer to as ''sampling with replacement'', we will end up with a probability of <math>\frac{1}{13}</math> for a queen draw. By the same argument, we obtain a probability for drawing a spade as <math>\frac{1}{4}</math>. The expression <math>P(A\cup B)</math> here can be translated as "the chance of drawing a queen or a spade". If we incorrectly assume that for this case <math>P(A\cup B) = P(A) + P(B)</math>, we can simply add our probabilities together for "the chance of drawing a queen or a spade" as <math>\frac{1}{13}+\frac{1}{4}</math>. If we were to gather some data experimentally, we would find that our results would differ from the prediction -- the probability observed would be slightly less than <math>\frac{1}{13}+\frac{1}{4}</math>. Why? Because we're counting the queen of spades twice in our expression, once as a spade, and again as a queen. We need to count it only once, as it can only be drawn with probability of <math>\frac{1}{52}</math>. [[Introduction to Statistics/Still confused?|Still confused?]] 

'''Proof:''' If <math>A</math> and <math>B</math> are not disjoint, we have to avoid the double counting problem by exactly specifying their union. 
<math>A \cup B = A \cup (B \backslash A) </math> so 
<math>P(A \cup B) = P(A \cup (B \backslash A)) </math> 
<math>A</math> and <math>B \backslash A</math> are disjoint sets. We can then use the definition of disjoint events from above to express our desired result: 
<math>P(A \cup B) = P(A) + P(B \backslash A)</math> 
We also know that 
<math>P(B \backslash A) = P(B) - P(B\cap A)</math> 
so 
<math> P(A \cup B) = P(A) + P(B) - P(B\cap A)</math> 
Whew! Our first proof. I hope that wasn't too [[dry]].

== Conditional Probability ==
Many events are conditional on the occurance of other events. Sometimes this coupling is weak. One event may become more or less probable depending on our knowledge that another event has occured. For instance, the probability that your friends and relatives will call asking for money is likely to be higher if you win the lottery. In my case, I don't think this probability would change.

Let's get formal for a second and remember our original definition of probability. 
<math>P(A) = \frac{N_{A}}{N}</math>
Consider an additional event <math>B</math>, and a situation where we are only interested in the probability of the occurrence of <math>A</math> when <math>B</math> occurs. A way at this probability is to perform a set of experiments (''trials'') and only record our results when the event <math>B</math> occurs. In other words 
<math> \frac{N_{A\cap B}}{N_{B}} </math> 
We can divide through on top and bottom by <math>N</math> the total number of trials to get <math>P(A\cap B)/P(B)</math>. We define this as ''''conditional probability'''': 
<math>P(A|B) = \frac{P(A\cap B)}{P(B)}</math> 
which when spoken, takes the sound "probability of <math>A</math> given <math>B</math>."

[[/A confusing problem/]]

=== Bayes' Law ===
'''THIS THEOREM IS NOT GOOD, THERE IS AN ERROR !!!!!!!''' 
'''JUST SEE THE RIGHT ONE AT THIS PAGE:''' 
http://en.wikipedia.org/wiki/Bayes'_theorem 
An important theorem in statistics is '''Bayes' Law''', which states that 
<math> P(B|A) = \frac{P(A|B)P(A)}{P(B)}</math>, 
It is easy to prove. We start with identical expressions for <math>P(A\cap B)</math>. 
We know that: <math> P(A\cap B) = P(B\cap A) </math>, 
<math> P(A\cap B) = \frac{P(A|B)}{P(B)}</math>, and 
<math> P(B\cap A) = \frac{P(B|A)}{P(A)}</math>. 
Since <math> P(A\cap B) = P(B\cap A) </math>, 
<math>\frac{P(A|B)}{P(B)} = \frac{P(B|A)}{P(A)}</math>. 
A Simple rearrangement of above line gives us '''Bayes' Law'''.

== Independence ==

Two events <math>A</math> and <math>B</math> are called ''independent'' if the occurence of one has absolutely no effect on the probability of the occurence of the other. Mathematically, this is expressed as: 
<math>P(A\cap B) = P(A)P(B)</math>. 

== Random Variables ==

It's usually possible to represent the outcome of experiments in terms of integers or real numbers. For instance, in the case of conducting a poll, it becomes a little cumbersome to present the outcomes of each individual respondant. Let's say we poll ten people for their voting preferences (Republican - R, or Democrat - D) in two different electorial districts. Our results might look like this: 
<math>\{RRRDRRDRRR\}</math> and <math>\{DDDDDRDDDD\}</math> 
But we're probably only interested in the overall breakdown in voting preference for each district. If we assign an integer value to each outcome, say 0 for Democrat and 1 for Republican, we can obtain a concise summary of voting preference by district simply by adding the results together.

== Discrete and Continuous Random Variables ==
There are two important subclasses of random variables: discrete random variable (DRV) and continuous random variable (CRV).
Discrete random variables take only countably many values. It means that we can list the set of all possible values that a discrete random variable can take, or in other words, the number of possible values in the set that the variable can take is finite. If the possible values that a DRV X can take are a0,a1,a2,...an, the probability that X takes each is p0=P(X=a0), p1=P(X=a1), p2=P(X=a2),...pn=P(X=an). All these probabilites are greater than or equal zero.

For continuous random variables, we cannot list all possible values that a continuous variable can take because the number of values it can take is extremely large. It means that there is no use to calculate the probability of each value separately because the probability that the variable takes a particular value is extremely small and can be considered zero P(X=x)=0).

== Distribution Functions ==

== Expectation Values ==

==See also==
* [[Topic:Statistics]]
* [[Introduction to research]]
* [[Statistical Economics]]
* [[Topic:Actuarial mathematics]]
* [[Introduction to Likelihood Theory]]
* [[Introduction to Classical Statistics]]
* [[Wikiversity:Statistics 202]]
* [[Introduction to probability and statistics]]
* [[Bayesian Statistics]]
* [[Statistics for Business Decisions]]
* [[Wikiversity:Statistics]]

[[Category:Mathematics]]
[[Category:Statistics]]
[[Category:Introductions]]

[[fr:Initiation à la statistique]]