AP Statistics Curriculum 2007 Laplace - Revision history

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

2020-03-03T19:23:24Z

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

IvoDinov: /* Related Distributions */ typos

2011-07-23T23:12:04Z

‎Related Distributions: typos

JayZzz at 22:42, 18 July 2011

2011-07-18T22:42:57Z

JayZzz at 22:42, 18 July 2011

2011-07-18T22:42:46Z

TracyTam: /* Related Distributions */

2011-07-11T21:44:56Z

‎Related Distributions

TracyTam: /* Related Distributions */

2011-07-11T21:41:02Z

‎Related Distributions

TracyTam: /* Laplace Distribution */

2011-07-11T21:23:27Z

‎Laplace Distribution

TracyTam: /* Related Distributions */

2011-07-11T21:22:50Z

‎Related Distributions

TracyTam: /* Laplace Distribution */

2011-07-11T21:13:58Z

‎Laplace Distribution

TracyTam: /* Applications */

2011-07-11T21:13:38Z

‎Applications

← Older revision		Revision as of 19:23, 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/AP_Statistics_Curriculum_2007_Laplace}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Laplace}}

@@ Line 45: / Line 45: @@
 *If <math>X\sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!</math>
-*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Exponential Exponential distribution]])
+*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[AP_Statistics_Curriculum_2007_Exponential |Exponential distribution]])
 *If <math>X \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math> and <math>Y \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math>, then <math>X-Y \sim \operatorname{Laplace}\left(0,\lambda\right)\!</math>
@@ Line 51: / Line 51: @@
 *If <math>X \sim \operatorname{Laplace}(\mu,\tfrac{1}{b})\!</math>, then <math>|X-\mu| \sim \operatorname{Exponential}(b)\!</math>
-* If <math>X_i \sim \operatorname{Normal}(0,1)\!</math>  for <math>i={1,2,3,4}\!</math> then <math>X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Std Normal distribution]])
+* If <math>X_i \sim \operatorname{Normal}(0,1)\!</math>  for <math>i={1,2,3,4}\!</math> then <math>X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> ([[AP_Statistics_Curriculum_2007_Normal_Std |Normal distribution]])
 * If <math>X_i \sim \operatorname{Normal}(0,1)\!</math> for <math>i={1,2,3,4}\!</math>, then <math>X_1 X_2 + X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math>
-* If <math>X_i \sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! </math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Chi-Square Chi-square distribution]])
+* If <math>X_i \sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! </math> ([[AP_Statistics_Curriculum_2007_Chi-Square |Chi-square distribution]])
-*If <math>X \sim \operatorname{Laplace}(\mu,b)</math> and <math>Y \sim \operatorname{Laplace}(\mu,b)</math>, then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Fisher_F F-distribution]])
+*If <math>X \sim \operatorname{Laplace}(\mu,b)</math> and <math>Y \sim \operatorname{Laplace}(\mu,b)</math>, then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[AP_Statistics_Curriculum_2007_Fisher_F |F-distribution]])
-*If <math>X \sim \operatorname{U}(0,1)\!</math> and <math>Y \sim \operatorname{U}(0,1)\!</math>,  then <math>\log{\tfrac{X}{Y}} \sim \operatorname{Laplace}(0,1)\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Uniform Uniform distribution]])
+*If <math>X \sim \operatorname{U}(0,1)\!</math> and <math>Y \sim \operatorname{U}(0,1)\!</math>,  then <math>\log{\tfrac{X}{Y}} \sim \operatorname{Laplace}(0,1)\!</math> ([[AP_Statistics_Curriculum_2007_Uniform |Uniform distribution]])
-*If <math>X \sim \operatorname{Exponential}(\lambda)\!</math> and  <math>Y \sim \operatorname{Bernoulli}(0.5)\!</math> independent of <math>X\!</math>, then <math> X(2Y-1) \sim \operatorname{Laplace} (0,\lambda^{-1}) \!</math>([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Distrib_Binomial Bernoulli distribution]])
+*If <math>X \sim \operatorname{Exponential}(\lambda)\!</math> and  <math>Y \sim \operatorname{Bernoulli}(0.5)\!</math> independent of <math>X\!</math>, then <math> X(2Y-1) \sim \operatorname{Laplace} (0,\lambda^{-1}) \!</math>([[AP_Statistics_Curriculum_2007_Distrib_Binomial |Bernoulli distribution]])
 *If <math>X_1 \sim \operatorname{Exponential}(\lambda_1)\!</math> and <math>X_2 \sim \operatorname{Exponential}(\lambda_2)\!</math> independent of <math>X_1\!</math>, then <math>\lambda_1 X_1-\lambda_2 X_2 \sim \operatorname{Laplace}\left(0,1\right)\,</math>

← Older revision		Revision as of 22:42, 18 July 2011
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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/~~AP_Statistics_Curriculum_2007_Chi-Square~~}}	+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Laplace}}

@@ Line 1: / Line 1: @@
 ===Laplace Distribution===
 '''Definition''': Laplace distribution is a distribution that is symmetrical and more “peaky” than a normal distribution. The dispersion of the data around the mean is higher than that of a normal distribution. Laplace distribution is also sometimes called the double exponential distribution.
@@ Line 86: / Line 88: @@
 The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html SOCR distributions]
 <center>[[Image:Laplace.jpg|600px]]</center>

@@ Line 43: / Line 43: @@
 *If <math>X\sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!</math>
-*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[Exponential distribution]])
+*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Exponential Exponential distribution]])
 *If <math>X \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math> and <math>Y \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math>, then <math>X-Y \sim \operatorname{Laplace}\left(0,\lambda\right)\!</math>
@@ Line 49: / Line 49: @@
 *If <math>X \sim \operatorname{Laplace}(\mu,\tfrac{1}{b})\!</math>, then <math>|X-\mu| \sim \operatorname{Exponential}(b)\!</math>
-* If <math>X_i \sim \operatorname{Normal}(0,1)\!</math>  for <math>i={1,2,3,4}\!</math> then <math>X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> ([[Normal distribution]])
+* If <math>X_i \sim \operatorname{Normal}(0,1)\!</math>  for <math>i={1,2,3,4}\!</math> then <math>X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Std Normal distribution]])
 * If <math>X_i \sim \operatorname{Normal}(0,1)\!</math> for <math>i={1,2,3,4}\!</math>, then <math>X_1 X_2 + X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math>
-* If <math>X_i \sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! </math> ([[Chi-square distribution]])
+* If <math>X_i \sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! </math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Chi-Square Chi-square distribution]])
-*If <math>X \sim \operatorname{Laplace}(\mu,b)</math> and <math>Y \sim \operatorname{Laplace}(\mu,b)</math>, then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[F-distribution]])
+*If <math>X \sim \operatorname{Laplace}(\mu,b)</math> and <math>Y \sim \operatorname{Laplace}(\mu,b)</math>, then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Fisher_F F-distribution]])
-*If <math>X \sim \operatorname{U}(0,1)\!</math> and <math>Y \sim \operatorname{U}(0,1)\!</math>,  then <math>\log{\tfrac{X}{Y}} \sim \operatorname{Laplace}(0,1)\!</math> ([[Uniform distribution]])
+*If <math>X \sim \operatorname{U}(0,1)\!</math> and <math>Y \sim \operatorname{U}(0,1)\!</math>,  then <math>\log{\tfrac{X}{Y}} \sim \operatorname{Laplace}(0,1)\!</math> ([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Uniform Uniform distribution]])
-*If <math>X \sim \operatorname{Exponential}(\lambda)\!</math> and  <math>Y \sim \operatorname{Bernoulli}(0.5)\!</math> independent of <math>X\!</math>, then <math> X(2Y-1) \sim \operatorname{Laplace} (0,\lambda^{-1}) \!</math>([[Bernoulli distribution]])
+*If <math>X \sim \operatorname{Exponential}(\lambda)\!</math> and  <math>Y \sim \operatorname{Bernoulli}(0.5)\!</math> independent of <math>X\!</math>, then <math> X(2Y-1) \sim \operatorname{Laplace} (0,\lambda^{-1}) \!</math>([[http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Distrib_Binomial Bernoulli distribution]])
 *If <math>X_1 \sim \operatorname{Exponential}(\lambda_1)\!</math> and <math>X_2 \sim \operatorname{Exponential}(\lambda_2)\!</math> independent of <math>X_1\!</math>, then <math>\lambda_1 X_1-\lambda_2 X_2 \sim \operatorname{Laplace}\left(0,1\right)\,</math>

@@ Line 2: / Line 2: @@
 '''Definition''': Laplace distribution is a distribution that is symmetrical and more “peaky” than a normal distribution. The dispersion of the data around the mean is higher than that of a normal distribution. Laplace distribution is also sometimes called the double exponential distribution.
-<br />'''Probability density function''': For X~Laplace(<math>\mu</math>,b), the Laplace probability density function is given by
+<br />'''Probability density function''': For <math>X\sim \operatorname{Laplace}(\mu,b)</math>, the Laplace probability density function is given by
 :<math>\frac{1}{2b}\exp(-\frac{|x-\mu|}{b})</math>

@@ Line 41: / Line 41: @@
 ===Related Distributions===
-*If <math>X\sim Laplace(\mu,b)\!</math>, then <math>kX+b\sim Laplace(k\mu+b,kb)\!</math>
+*If <math>X\sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!</math>
-*If <math>X \sim Laplace(0,b)\!</math>, then <math>|X| \sim Exponential(\tfrac{1}{b})\!</math> ([[exponential distribution]])
+*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[Exponential distribution]])
 ===Applications===

← Older revision		Revision as of 21:13, 11 July 2011
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	*Insurance claims		*Insurance claims
	*Structural changes in switching-regime model and Kalman filter		*Structural changes in switching-regime model and Kalman filter
		+
		+	===Example===
		+	Suppose that the return of a certain stock has a Laplace distribution with <font size=3><math>\mu=5</math></font> and <font size=3><math>b=2</math></font>. Compute the probability that the stock will have a return between 6 and 10.
		+
		+	We can compute this as follows:
		+
		+	:<math>P(6 \le X\le 10)=\sum_{x=6}^{10}\frac{1}{2\times 2}\exp(-\frac{\|x-5\|}{2})=0.262223</math>
		+
		+	The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html SOCR distributions]
		+	<center>[[Image:Laplace.jpg\|600px]]</center>