Difference between revisions of "Documentation/How Tos/Calc: BETADIST function"
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== 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. | |
| − | + | 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>'''a'''</tt> and <tt>'''b'''</tt> are optional parameters which default (if omitted) to <tt>'''0'''</tt> and <tt>'''1'''</tt>. | |
| − | <tt>''' | + | |
| − | + | ||
| − | === See also | + | <tt>'''cumulative'''</tt> is an optional 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, than the density function has a pole at x = a. | ||
| + | # If β < 1, than 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 === | ||
| + | [[Image:BetaDistributionDensityLocationScale.png|Graphs of Beta distribution density functions using parameter a and b]] | ||
| + | |||
| + | <tt>'''BETADIST(1;5;3;-2;4;FALSE())</tt> | ||
| + | : returns approximately 0.273 | ||
| + | |||
| + | [[Image:BetaDistributionDensityNormal.png|Graphs of Beta distribution density functions with borders 0 and 1]] | ||
| + | |||
| + | <tt>'''BETADIST(0.2;0.7;4;0;1;FALSE())</tt> | ||
| + | : returns approximately 1.644 | ||
| + | |||
| + | [[Image:BetaDistributionDensitySpecial.png|Graphs of Beta distribution density functions with alpha=1]] | ||
| + | |||
| + | <tt>'''BETADIST(1;1;0.5;0;1;FALSE())</tt> | ||
| + | : returns ''illegal argument'' because there is a pole | ||
| + | |||
| + | <tt>'''BETADIST(1.1;1;0.5;0;1;FALSE())</tt> | ||
| + | : returns 0 | ||
| + | |||
| + | === See also === | ||
[[Documentation/How_Tos/Calc: BETAINV function|'''BETAINV''']] | [[Documentation/How_Tos/Calc: BETAINV function|'''BETAINV''']] | ||
Revision as of 18:47, 6 December 2008
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.
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 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, than the density function has a pole at x = a.
- If β < 1, than 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())
- returns approximately 0.273
BETADIST(0.2;0.7;4;0;1;FALSE())
- returns approximately 1.644
BETADIST(1;1;0.5;0;1;FALSE())
- returns illegal argument because there is a pole
BETADIST(1.1;1;0.5;0;1;FALSE())
- returns 0
See also
Functions listed alphabetically, Functions listed by category
