AP Statistics Curriculum 2007 Beta

(Difference between revisions)
 Revision as of 20:54, 11 July 2011 (view source)TracyTam (Talk | contribs) (→Beta Distribution)← Older edit Current revision as of 22:35, 18 July 2011 (view source)JayZzz (Talk | contribs) (6 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. -
'''Probability density function''': For $X\sim Beta(\alpha,\beta)\!$, the Beta probability density function is given by +
'''Probability density function''': For $X\sim \operatorname{Beta}(\alpha,\beta)\!$, the Beta probability density function is given by :$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}$ :$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}$ Line 9: Line 11: *$\alpha$ is a positive shape parameter *$\alpha$ is a positive shape parameter *$\beta$ is a positive shape parameter *$\beta$ is a positive shape parameter - *$\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt$ or + *$\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt$ or
$\textstyle\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$, where $\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)$ - :$\textstyle\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$, where $\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)$ + *x is a random variable *x is a random variable Line 18: Line 19: where where - *$\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt$ + *$\textstyle\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt$ - *$\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt$ + *$\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt$
'''Moment generating function''': The Beta moment-generating function is
'''Moment generating function''': The Beta moment-generating function is Line 37: 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 $\alpha=2$ and $\beta=5$. 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 48: 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]
[[Image:Beta.jpg|600px]]
[[Image:Beta.jpg|600px]]
+ +

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