Beta distribution
In theory of probability and in statistics, the distribution beta is a family of continuous distributions of probability defined in the interval (0,1){displaystyle (0.1)} parameterized by two positive parameters in form, denotated by α α {displaystyle alpha } and β β {displaystyle beta }, which appear as exponents of the random variable and control the form of the distribution.
The generalization of this distribution to several variables is known as the Dirichlet distribution.
Definition
Notation
If a continuous random variable X{displaystyle X} She's got one. distribution beta with parameters 0}" xmlns="http://www.w3.org/1998/Math/MathML">α α ,β β ▪0{displaystyle alphabeta /2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f650e33744628e414c5e7bc24ecf964e3a39554f" style="vertical-align: -0.671ex; width:8.114ex; height:2.509ex;"/> Then we'll write X♥ ♥ B(α α ,β β ){displaystyle Xsim mathrm {B} (alphabeta)}.
Other notations for the beta distribution used are X♥ ♥ Beta (α α ,β β ){displaystyle Xsim operatorname {Beta} (alphabeta)}, X♥ ♥ Be(α α ,β β ){displaystyle Xsim {mathcal {Be}(alphabeta)} or X♥ ♥ β β α α ,β β {displaystyle Xsim beta _{alphabeta }}.
Density function
The density function X{displaystyle X} That's it.
- fX(x)=xα α − − 1(1− − x)β β − − 1B(α α ,β β ){displaystyle f_{X}(x)={frac {x^{alpha}(1-x)^{beta -1}{mathrm {B} (alphabeta)}}}}}}}}}{mathrm {B}
for values <math alttext="{displaystyle 0<x0.x.1{displaystyle 0 ingredientx<img alt="{displaystyle 0<x where B(α α ,β β ){displaystyle mathrm {B} (alphabeta)} is beta function and is defined for 0}" xmlns="http://www.w3.org/1998/Math/MathML">α α ,β β ▪0{displaystyle alphabeta /2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f650e33744628e414c5e7bc24ecf964e3a39554f" style="vertical-align: -0.671ex; width:8.114ex; height:2.509ex;"/> Like
- B(α α ,β β )=∫ ∫ 01xα α − − 1(1− − x)β β − − 1dx{displaystyle mathrm {B} (alphabeta)=int _{0}^{1}x^{alpha-1}(1-x)^{beta -1}dx}
and some of the properties it satisfies are:
- B(α α ,β β )=B(β β ,α α ){displaystyle mathrm {B} (alphabeta)=mathrm {B} (betaalpha)}
- B(α α ,β β )=Interpreter Interpreter (α α )Interpreter Interpreter (β β )Interpreter Interpreter (α α +β β ){displaystyle mathrm {B} (alphabeta)={frac {Gamma (alpha)Gamma (beta)}{Gamma (alpha +beta)}}}}}}}
Distribution function
The distribution function X{displaystyle X} That's it.
- FX(x)=B(x;α α ,β β )B(α α ,β β )=Ix(α α ,β β ){displaystyle F_{X}(x)={frac {mathrm {B} (x;alphabeta)}{mathrm {B} (alphabeta)}}}{I_{x}(alphabeta)}}
where B(x;α α ,β β ){displaystyle mathrm {B} (x;alphabeta)} is the incomplete beta function and Ix(α α ,β β ){displaystyle I_{x}(alphabeta)} is the regular incomplete beta function.
Properties
Yeah. X♥ ♥ B(α α ,β β ){displaystyle Xsim mathrm {B} (alphabeta)} then the random variable X{displaystyle X} satisfies some properties.
Media
The mean of the random variable X{displaystyle X} That's it.
- E[chuckles]X]=α α α α +β β {displaystyle {text{E}}[X]={frac {alpha }{alpha +beta }}}}
![{displaystyle {text{E}}[X]={frac {alpha }{alpha +beta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8d12ff6bcad642fa716ef1dc34ae4d4dd01650)
Variance
Variance of the random variable X{displaystyle X} That's it.
- Var(X)=α α β β (α α +β β +1)(α α +β β )2{displaystyle {text{Var}}(X)={frac {alpha beta }{alpha +beta +1)(alpha +beta)^{2}}}}}}}}.
Fashion
The fashion of the random variable X{displaystyle X} That's it.
- α α − − 1α α +β β − − 2{displaystyle {frac {alpha-1}{alpha +beta-2}}
for values 1}" xmlns="http://www.w3.org/1998/Math/MathML">α α ,β β ▪1{displaystyle alphabeta }1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3f33fc553c096bb6e12987a13ab58edef863b6" style="vertical-align: -0.671ex; width:8.114ex; height:2.509ex;"/>.
Moments
The n{displaystyle n}- That's a moment. X{displaystyle X} That's it.
- E[chuckles]Xn]=B(α α +n,β β )B(α α +β β )= r=0n− − 1α α +rα α +β β +r=α α (α α +1) (α α +n− − 1)(α α +β β )(α α +β β +1) (α α +β β +n− − 1){displaystyle {begin{aligned}{text{E}}[X^{n}{frac {mathrm {B} (alpha +n,beta)}{mathrm {Balpha}}}}{alphat +beta
![{displaystyle {begin{aligned}{text{E}}[X^{n}]&={frac {mathrm {B} (alpha +n,beta)}{mathrm {B} (alpha +beta)}}\&=prod _{r=0}^{n-1}{frac {alpha +r}{alpha +beta +r}}\&={frac {alpha (alpha +1)cdots (alpha +n-1)}{(alpha +beta)(alpha +beta +1)cdots (alpha +beta +n-1)}}end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7119244b8e8f522b61b45f07dc0ce9054c7838)
for n한 한 N{displaystyle nin mathbb {N} }.
Moment generating function
The time-generating function of the random variable X{displaystyle X} is given by
- MX(t)=␡ ␡ n=0∞ ∞ tnn!B(α α +n,β β )B(α α ,β β )=1+␡ ␡ n=1∞ ∞ ( r=0n− − 1α α +rα α +β β +r)tnn!{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF00}{cHFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF00}{cHFFFF00}{cHFFFFFF00}{cHFFFFFFFFFF00}{cHFFFFFFFFFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cHFFFFFFFF00}{c}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFF00}{cH00}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
Geometric mean
The logarithm of the geometric media GX{displaystyle G_{X}} of a distribution with random variable X{displaystyle X} is the arithmetic average ln (X){displaystyle ln(X)} or equivalent, its expected value:
- ln GX=E [chuckles]ln X]{displaystyle ln G_{X}=operatorname {E} [ln X]
![{displaystyle ln G_{X}=operatorname {E} [ln X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b67cb73b90bc0e09ba41003b44f84b6e1d3feb)
For a beta distribution:
- E [chuckles]ln X]=∫ ∫ 01ln xfX(x)dx=∫ ∫ 01ln xxα α − − 1(1− − x)β β − − 1B(α α ,β β )dx=1B(α α ,β β )∫ ∫ 01▪ ▪ xα α − − 1(1− − x)β β − − 1▪ ▪ α α dx=1B(α α ,β β )▪ ▪ ▪ ▪ α α ∫ ∫ 01xα α − − 1(1− − x)β β − − 1dx=1B(α α ,β β )▪ ▪ B(α α ,β β )▪ ▪ α α =▪ ▪ ln B(α α ,β β )▪ ▪ α α =▪ ▪ ln Interpreter Interpreter (α α )▪ ▪ α α − − ▪ ▪ ln Interpreter Interpreter (α α +β β )▪ ▪ α α =END END (α α )− − END END (α α +β β ){cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFF}{cHFFFFFFFFFF}{cHFFFFFFFFFFFF}{cHFFFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{
![{displaystyle {begin{aligned}operatorname {E} [ln X]&=int _{0}^{1}ln xf_{X}(x)dx\&=int _{0}^{1}ln x;{frac {x^{alpha -1}(1-x)^{beta -1}}{mathrm {B} (alphabeta)}}dx\&={frac {1}{mathrm {B} (alphabeta)}}int _{0}^{1}{frac {partial x^{alpha -1}(1-x)^{beta -1}}{partial alpha }};dx\&={frac {1}{mathrm {B} (alphabeta)}}{frac {partial }{partial alpha }}int _{0}^{1}x^{alpha -1}(1-x)^{beta -1}dx\&={frac {1}{mathrm {B} (alphabeta)}}{frac {partial mathrm {B} (alphabeta)}{partial alpha }}\&={frac {partial ln mathrm {B} (alphabeta)}{partial alpha }}\&={frac {partial ln Gamma (alpha)}{partial alpha }}-{frac {partial ln Gamma (alpha +beta)}{partial alpha }}\&=psi (alpha)-psi (alpha +beta)end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad100ac91c363b10f11f43b63896a6c614ca0935)
where END END {displaystyle psi } It's the digomma function.
Related distributions
Transformations
- Yeah. X♥ ♥ B(α α ,β β ){displaystyle Xsim mathrm {B} (alphabeta)} then. 1− − X♥ ♥ B(β β ,α α ){displaystyle 1-Xsim mathrm {B} (betaalpha)}.
- Yeah. X♥ ♥ B(α α ,β β ){displaystyle Xsim mathrm {B} (alphabeta)} then. X1− − X♥ ♥ β β ♫(α α ,β β ){displaystyle {frac {X}{1-X}}sim beta '(alphabeta)}the beta distribution of second order.
- Yeah. X♥ ♥ B(n2,m2){displaystyle Xsim mathrm {B} left({frac {n}{2}}}}},{frac {m}{2}}right)} then. mXn(1− − X)♥ ♥ Fn,m{displaystyle {frac {mX}{n(1-X)}}sim F_{n,m}}.
- Yeah. X♥ ♥ B(α α ,1){displaystyle Xsim mathrm {B} (alpha1)} then. − − ln (X)♥ ♥ Exponencial (α α ){displaystyle -ln(X)sim operatorname {Exponencial} (alpha)}}.
Particular cases
- Yeah. X♥ ♥ B(1,1){displaystyle Xsim mathrm {B} (1,1)} then. X♥ ♥ U (0,1){displaystyle Xsim operatorname {U} (0.1)}.
- limn→ → ∞ ∞ nB(1,n)=Exponencial (1){displaystyle lim _{nto infty }nmathrm {B} (1,n)=operatorname {Exponencial} (1)}.
- limn→ → ∞ ∞ nB(k,n)=Interpreter Interpreter (k,1){displaystyle lim _{nto infty }nmathrm {B} (k,n)=Gamma (k,1)}.
- A particler case of the Beta Distribution is the PERT Distribution that takes three parameters: Optimist, more frequent and pessimistic.
