AP Statistics Curriculum 2007 Bayesian Prelim - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T18:41:46Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2010-12-20T18:32:08Z

‎Example: fixed a calculation typo (0.193 --> 0.3231293)

2010-06-28T21:13:34Z

2010-06-28T17:44:53Z

2009-10-22T22:33:06Z

2009-10-22T22:32:42Z

minor formatting and added examples

2009-07-23T20:52:27Z

2009-07-23T20:29:32Z

2009-07-23T20:26:48Z

2009-07-23T20:25:52Z

← Older revision		Revision as of 18:41, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Prelim}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Prelim}}

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 Denote D = {the test person has the disease}, <math>D^c</math> = {the test person does not have the disease} and  T = {the test result is positive}. Then
 <center><math>P(D | T) = {P(T | D) P(D) \over P(T)} = {P(T | D) P(D) \over P(T|D)P(D) + P(T|D^c)P(D^c)}=</math>
-<math>={0.95\times 0.005 \over {0.95\times 0.005 +0.01\times 0.995}}=0.193.</math></center>
+<math>={0.95\times 0.005 \over {0.95\times 0.005 +0.01\times 0.995}}=0.3231293.</math></center>
 ===Bayesian Statstics===

@@ Line 2: / Line 2: @@
 ===Introduction===
-Bayes theorem, or "Bayes Rule" can be stated succinctly by the equality
+Bayes Theorem, or "Bayes Rule" can be stated succinctly by the equality
 : <math>P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}</math>
@@ Line 13: / Line 13: @@
 ===Example===
-Suppose a Laboratory blood test is used as evidence for a disease. Assume P(positive Test| Disease) = 0.95, P(positive Test| no Disease)=0.01 and P(Disease) = 0.005. Find P(Disease|positive Test)=?
+Suppose a laboratory blood test is used as evidence for a disease. Assume P(positive Test| Disease) = 0.95, P(positive Test| no Disease)=0.01 and P(Disease) = 0.005. Find P(Disease|positive Test)=?
 Denote D = {the test person has the disease}, <math>D^c</math> = {the test person does not have the disease} and  T = {the test result is positive}. Then
@@ Line 19: / Line 19: @@
 <math>={0.95\times 0.005 \over {0.95\times 0.005 +0.01\times 0.995}}=0.193.</math></center>
-===Bayesian statstics===
+===Bayesian Statstics===
 What is commonly called '''Bayesian Statistics''' is a very special application of Bayes Theorem.

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 What is commonly called '''Bayesian Statistics''' is a very special application of Bayes Theorem.
-We will examine a number of examples in this Chapter, but to illustrate generally, imagine that '''x''' is a fixed collection of data that has been realized from under some known density, <math>f(X)</math>, that takes a parameter, <math>\mu</math>, whose value is not certainly known.
+We will examine a number of examples in this Chapter, but to illustrate generally, imagine that '''x''' is a fixed collection of data that has been realized from some known density, <math>f(X)</math>, that takes a parameter, <math>\mu</math>, whose value is not certainly known.
 Using Bayes Theorem we may write

← Older revision		Revision as of 22:33, 22 October 2009
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=~~AP_Statistics_Curriculum_2007_Contingency_Fit~~}}	+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Prelim}}

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-'''Bayes Theorem'''
+==[[EBook | Probability and Statistics Ebook]] - Bayes Theorem==
 Bayes theorem, or "Bayes Rule" can be stated succinctly by the equality
-<math>P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}</math>
+: <math>P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}</math>
 In words, "the probability of event A occurring given that event B occurred is equal to the probability of event B occurring given that event A occurred times the probability of event A occurring divided by the probability that event B occurs."
@@ Line 11: / Line 12: @@
 <math>f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }</math>
 What is commonly called '''Bayesian Statistics''' is a very special application of Bayes Theorem.
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 Using Bayes Theorem we may write
-<math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math>
+: <math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math>
 In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter, <math>\mu</math>.
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 <math>f(\mathbf{x}) = \int_{\mu} f( \mathbf{x} \cap \mu) d\mu =  \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math>

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 In words, "the probability of event A occurring given that event B occurred is equal to the probability of event B occurring given that event A occurred times the probability of event A occurring divided by the probability that event B occurs."
-Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. So, if <math>X</math> and <math>Y</math> are random variables, and <math>f(\cdot)</math> is a density or likelihood, we can say
+Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. Let's call <math>f(\star)</math> the density (or in some cases, the likelihood) defined by the random process <math>\star</math>.  If <math>X</math> and <math>Y</math> are random variables, we can say
 <math>f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }</math>