Difference between revisions of "Documentation/How Tos/Calc: BETADIST function"
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− | __NOTOC__ | + | {{DISPLAYTITLE:BETADIST function}} |
+ | {{Documentation/CalcFunc StatisticalTOC | ||
+ | |ShowPrevNext=block | ||
+ | |PrevPage=Documentation/How_Tos/Calc:_B_function | ||
+ | |NextPage=Documentation/How_Tos/Calc:_BETAINV_function | ||
+ | }}__NOTOC__ | ||
== BETADIST == | == BETADIST == | ||
− | Calculates the cumulative distribution function of a beta distribution. | + | Calculates the cumulative distribution function or the probability density function of a beta distribution. |
− | === Syntax | + | === Syntax === |
− | <tt>'''BETADIST( | + | <tt>'''BETADIST(x; α; β; a; b; cumulative)'''</tt> |
− | |||
− | + | The beta distribution is a family of continuous probability distributions, where <tt>'''α'''</tt> and <tt>'''β'''</tt> are parameters controlling the shape of the distribution. | |
− | + | <tt>'''x'''</tt> is the number, at which you will evaluate the Beta distribution. | |
− | + | The parameters <tt>'''a'''</tt> and <tt>'''b'''</tt> are lower and upper bounds of the distribution. You can interpret the value a as ''location'' and the value b−a as ''scale''. | |
− | <tt>''' | ||
− | |||
− | + | <tt>'''a'''</tt> and <tt>'''b'''</tt> are optional parameters which default (if omitted) to <tt>'''0'''</tt> and <tt>'''1'''</tt>. | |
− | |||
− | + | <tt>'''cumulative'''</tt> is an optional, logical parameter which defaults to TRUE() if omitted. | |
− | [[Documentation/How_Tos/Calc: Functions listed alphabetically| | + | Before OOo version 3.1 the parameter ''cumulative'' doesn't exist and only the cumulative probability function is calculated. |
− | [[Documentation/How_Tos/Calc: Functions listed by category| | + | |
+ | Constraints: | ||
+ | # α > 0 , β > 0 , a < b | ||
+ | # If α < 1, then the density function has a pole at x = a. | ||
+ | # If β < 1, then the density function has a pole at x = b. | ||
+ | |||
+ | === Semantic === | ||
+ | |||
+ | For <tt>'''cumulative = FALSE()'''</tt> the function BETADIST calculates the probability density function (besides the constraints given above): | ||
+ | :<math>f(x)= | ||
+ | \begin{cases} | ||
+ | 0, & \textrm{if}\; x < a \or x > b \\ | ||
+ | \displaystyle \frac {\Gamma(\alpha + \beta)} {\Gamma(\alpha) \Gamma(\beta)} \cdot \left( \frac {x-a} {b-a} \right)^{\alpha-1}\cdot \left(1- \frac{x-a}{b-a} \right)^{\beta-1} \cdot \frac{1}{b-a}, & \textrm{if}\; a \le x \le b | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | For <tt>'''cumulative = TRUE()'''</tt> the function BETADIST calculates the cumulative distribution function: | ||
+ | :<math>F(x)= | ||
+ | \begin{cases} | ||
+ | 0, & \textrm{if}\; x<a \\ | ||
+ | 1, & \textrm{if}\; x>b \\ | ||
+ | \displaystyle \int_a^x \frac {\Gamma(\alpha + \beta)} {\Gamma(\alpha) \Gamma(\beta)} \cdot \left( \frac {x-a} {b-a} \right)^{\alpha-1}\cdot \left(1- \frac{x-a}{b-a} \right)^{\beta-1} \mathrm{dt}, & \textrm{if}\; a \le x \le b | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | Notice, that | ||
+ | :<math>\displaystyle F(x)=I_z(\alpha,\beta)</math> | ||
+ | where | ||
+ | :<math>\displaystyle z = \frac {x-a}{b-a}</math> | ||
+ | and <math>\scriptstyle I_z</math> is the regularized incomplete Beta function. | ||
+ | |||
+ | === Example === | ||
+ | {| | ||
+ | |<tt>'''BETADIST(1;5;3;-2;4;FALSE())</tt> | ||
+ | : returns approximately 0.273 | ||
+ | |[[Image:BetaDistributionDensityLocationScale.png|Graphs of Beta distribution density functions using parameter a and b]] | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |<tt>'''BETADIST(0.2;0.7;4;0;1;FALSE())</tt> | ||
+ | : returns approximately 1.644 | ||
+ | |[[Image:BetaDistributionDensityNormal.png|Graphs of Beta distribution density functions with borders 0 and 1]] | ||
+ | |} | ||
+ | |||
+ | |||
+ | {| | ||
+ | |<tt>'''BETADIST(1;1;0.5;0;1;FALSE())</tt> | ||
+ | : returns ''Invalid argument'' because there is a pole | ||
+ | |||
+ | <tt>'''BETADIST(1.1;1;0.5;0;1;FALSE())</tt> | ||
+ | : returns 0 | ||
+ | |[[Image:BetaDistributionDensitySpecial.png|Graphs of Beta distribution density functions with alpha=1]] | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |<tt>'''BETADIST(1;5;3;-2;4;TRUE())</tt> | ||
+ | : returns approximately 0.227 | ||
+ | call before version 3.1: <tt>'''BETADIST(1;5;3;-2;4)</tt> | ||
+ | |[[Image:BetaDistributionCumulativeLocationScale.png|Graphs of Beta distribution cumulative functions using parameter a and b]] | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |<tt>'''BETADIST(0.2;0.7;4;0;1;TRUE())</tt> | ||
+ | : returns approximately 0.718 | ||
+ | call before version 3.1: <tt>'''BETADIST(0.2;0.7:4;0;1)</tt> | ||
+ | |||
+ | other valid call: <tt>'''BETADIST(0.2;0.7;4)</tt> | ||
+ | |[[Image:BetaDistributionCumulativeNormal.png|Graphs of Beta distribution cumulative functions with borders 0 and 1]] | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |<tt>'''BETADIST(1.1;1;0.5;0;1;TRUE())</tt> | ||
+ | : returns 1 | ||
+ | before version 3.1: <tt>'''BETADIST(1.1;1;0.5;0;1)</tt> | ||
+ | : returns ''Invalid argument'' | ||
+ | |[[Image:BetaDistributionCumulativeSpecial.png|Graphs of Beta distribution cumulative functions with alpha=1]] | ||
+ | |} | ||
+ | |||
+ | {{SeeAlso|EN| | ||
+ | * [[Documentation/How_Tos/Calc: BETAINV function|BETAINV]] | ||
+ | |||
+ | * [[Documentation/How_Tos/Calc: Statistical functions|Statistical functions]] | ||
+ | |||
+ | * [[Documentation/How_Tos/Calc: Functions listed alphabetically|Functions listed alphabetically]] | ||
+ | * [[Documentation/How_Tos/Calc: Functions listed by category|Functions listed by category]] | ||
+ | }} | ||
+ | [[Category: Documentation/Reference/Calc/Statistical functions]] |
Latest revision as of 13:39, 2 February 2024
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BETADIST
Calculates the cumulative distribution function or the probability density function of a beta distribution.
Syntax
BETADIST(x; α; β; a; b; cumulative)
The beta distribution is a family of continuous probability distributions, where α and β are parameters controlling the shape of the distribution.
x is the number, at which you will evaluate the Beta distribution.
The parameters a and b are lower and upper bounds of the distribution. You can interpret the value a as location and the value b−a as scale.
a and b are optional parameters which default (if omitted) to 0 and 1.
cumulative is an optional, logical parameter which defaults to TRUE() if omitted.
Before OOo version 3.1 the parameter cumulative doesn't exist and only the cumulative probability function is calculated.
Constraints:
- α > 0 , β > 0 , a < b
- If α < 1, then the density function has a pole at x = a.
- If β < 1, then the density function has a pole at x = b.
Semantic
For cumulative = FALSE() the function BETADIST calculates the probability density function (besides the constraints given above):
For cumulative = TRUE() the function BETADIST calculates the cumulative distribution function:
Notice, that
where
and is the regularized incomplete Beta function.
Example
BETADIST(1;5;3;-2;4;FALSE())
|
BETADIST(0.2;0.7;4;0;1;FALSE())
|
BETADIST(1;1;0.5;0;1;FALSE())
BETADIST(1.1;1;0.5;0;1;FALSE())
|
BETADIST(1;5;3;-2;4;TRUE())
call before version 3.1: BETADIST(1;5;3;-2;4) |
BETADIST(0.2;0.7;4;0;1;TRUE())
call before version 3.1: BETADIST(0.2;0.7:4;0;1) other valid call: BETADIST(0.2;0.7;4) |
BETADIST(1.1;1;0.5;0;1;TRUE())
before version 3.1: BETADIST(1.1;1;0.5;0;1)
|
{{#switch:EN
| RU = Смотрите также:
| UA = Дивіться також:
| EN = See Also
| PT = Ver também
| #default = See Also
}}