AP Statistics Curriculum 2007 Beta

From Socr

(Difference between revisions)
Jump to: navigation, search
(Created page with '===Beta Distribution=== '''Definition''': Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and m…')
 
(7 intermediate revisions not shown)
Line 1: Line 1:
 +
==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Beta Distribution==
 +
===Beta Distribution===
===Beta Distribution===
'''Definition''': Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value.  
'''Definition''': Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value.  
-
<br />'''Probability density function''': For X~Beta(<math>\alpha,\beta</math>), the Beta probability density function is given by
+
<br />'''Probability density function''': For <math>X\sim \operatorname{Beta}(\alpha,\beta)\!</math>, the Beta probability density function is given by
:<math>\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}</math>
:<math>\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}</math>
where
where
-
*<math>\alpha</math> is a positive shape parameter
+
*<font size="3"><math>\alpha</math></font> is a positive shape parameter
-
*<math>\beta</math> is a positive shape parameter
+
*<font size="3"><math>\beta</math></font> is a positive shape parameter
-
*<math>\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or
+
*<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or <br /><math>\textstyle\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}</math>, where <math>\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)</math>
-
:<math>\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}</math>, where <math>\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)</math>
+
*x is a random variable
*x is a random variable
Line 18: Line 19:
where
where
-
*<math>\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt</math>
+
*<math>\textstyle\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt</math>
-
*<math>\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math>
+
*<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math>
<br />'''Moment generating function''': The Beta moment-generating function is
<br />'''Moment generating function''': The Beta moment-generating function is
Line 32: Line 33:
:<math>Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}</math>
:<math>Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}</math>
-
 
===Applications===
===Applications===
Line 38: Line 38:
*The time it takes to complete a task
*The time it takes to complete a task
*The proportion of defective items in a shipment
*The proportion of defective items in a shipment
-
 
===Example===
===Example===
-
Suppose that DVDs in a certain shipment are defective with a Beta distribution with α=2 and β=5. Compute the probability that the shipment has 20% to 30% defective DVDs.
+
Suppose that DVDs in a certain shipment are defective with a Beta distribution with <font size="3"><math>\alpha=2</math></font> and <font size="3"><math>\beta=5</math></font>. Compute the probability that the shipment has 20% to 30% defective DVDs.
We can compute this as follows:   
We can compute this as follows:   
Line 49: Line 48:
The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html SOCR distributions]
The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html SOCR distributions]
<center>[[Image:Beta.jpg|600px]]</center>
<center>[[Image:Beta.jpg|600px]]</center>
 +
 +
<hr>
 +
* SOCR Home page: http://www.socr.ucla.edu
 +
 +
{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Beta}}

Current revision as of 22:33, 18 July 2011

Contents

General Advance-Placement (AP) Statistics Curriculum - Beta Distribution

Beta Distribution

Definition: Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value.


Probability density function: For X\sim \operatorname{Beta}(\alpha,\beta)\!, the Beta probability density function is given by

\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}

where

  • α is a positive shape parameter
  • β is a positive shape parameter
  • \textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt or
    \textstyle\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}, where \Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)
  • x is a random variable


Cumulative density function: Beta cumulative distribution function is given by

\frac{\Beta_x(\alpha,\beta)}{\Beta(\alpha,\beta)}

where

  • \textstyle\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt
  • \textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt


Moment generating function: The Beta moment-generating function is

M(t)=1+\sum_{k=1}^\infty (\prod_{r=0}^{k-1}\frac{\alpha+r}{\alpha+\beta+r})\frac{t^k}{k!}


Expectation: The expected value of a Beta distributed random variable x is

E(X)=\frac{\alpha}{\alpha+\beta}


Variance: The Beta variance is

Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

Applications

The Beta distribution is used in a range of disciplines including rule of succession, Bayesian statistics, and task duration modeling. Examples of events that may be modeled by Beta distribution include:

  • The time it takes to complete a task
  • The proportion of defective items in a shipment

Example

Suppose that DVDs in a certain shipment are defective with a Beta distribution with α = 2 and β = 5. Compute the probability that the shipment has 20% to 30% defective DVDs.

We can compute this as follows:

P(0.2\le X\le 0.3)=\sum_{x=0.2}^{0.3}\frac{x^{2-1}(1-x)^{5-1}}{\Beta(2,5)}=0.235185

The figure below shows this result using SOCR distributions




Translate this page:

(default)

Deutsch

Español

Français

Italiano

Português

日本語

България

الامارات العربية المتحدة

Suomi

इस भाषा में

Norge

한국어

中文

繁体中文

Русский

Nederlands

Ελληνικά

Hrvatska

Česká republika

Danmark

Polska

România

Sverige

Personal tools