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:

1. B(α α ,β β )=B(β β ,α α ){displaystyle mathrm {B} (alphabeta)=mathrm {B} (betaalpha)}
2. 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 }}}}

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

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]

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}{

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.