AP Statistics Curriculum 2007 Beta

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(Beta Distribution)
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'''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.  
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<br />'''Probability density function''': For <math>X\sim Beta(\alpha,\beta)\!</math>, the Beta probability density function is given by
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<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>
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*<font size="3"><math>\alpha</math></font> is a positive shape parameter
*<font size="3"><math>\alpha</math></font> is a positive shape parameter
*<font size="3"><math>\beta</math></font> is a positive shape parameter
*<font size="3"><math>\beta</math></font> is a positive shape parameter
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*<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or
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*<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or <br />
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:<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>
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<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>
*x is a random variable
*x is a random variable

Revision as of 21:17, 11 July 2011

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

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