UFABC - Mecânica Estatística Curso 2018.2 - Prof. Germán Lugones

CAPITULO 2 Variáveis Aleatórias

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Introdução

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H(pi ,qi) QN(V, T) F(T, V, N)

Mecânica Mecânica Estatística Termodinâmica

Hamiltoniano Função de partição Relação fundamental: entropia ou energia interna, ou potencial termodinâmico.

Para discutir as propriedades de equilíbrio de um sistema macroscópico, não é necessário o pleno conhecimento do comportamento microscópico de suas partículas constituintes.

Tudo o que é necessário é a probabilidade de que as partículas estejam num estado microscópico particular.

A mecânica estatística é, portanto, uma descrição inerentemente probabilística do sistema e a familiaridade com as manipulações das probabilidades é um pré-requisito importante.

O objetivo deste capítulo é revisar, de maneira informal, alguns resultados importantes na teoria da probabilidade.

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Variável aleatória: é uma grandeza X que pode assumir diferentes valores numéricos x, definidos para eventos de um espaço amostral !:

! ≡ {x1 , x2 , · · ·} O resultado (valor) depende de fatores aleatórios. Por exemplo, o resultado do lançamento de um dado pode dar qualquer número entre 1 e 6; !dado ≡ {1, 2, 3, 4, 5, 6}

Embora possamos conhecer os seus possíveis resultados, o resultado em si depende de fatores de sorte.

As variáveis aleatórias podem ser discretas ou contínuas: • discreto: dado → !dado ≡ {1, 2, 3, 4, 5, 6} • contínuo: a velocidade de uma partícula de um gás,

!v ≡ { −∞< vx, vy, vz < +∞} !4

Variáveis aleatórias

Um evento é qualquer subconjunto de resultados E, contido no espaço amostral, (E ⊂ !), ao qual pode ser atribuída uma probabilidade p(E).

Por exemplo: para um dado honesto, temos: pdado({1}) = 1/6 pdado({1,3}) = 1/3

As probabilidades devem satisfazer as seguintes propriedades:

• Positividade: p(E) ≥ 0; todas as probabilidades devem ser reais e não negativas.

• Aditividade: p(A ou B) = p(A) + p(B) , se A e B são eventos desconectados entre si.

• Normalização: p(!) = 1.

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Do ponto de vista prático, gostaríamos de saber como atribuir valores de probabilidade aos diferentes resultados possíveis.

Para isso temos duas estratégias possíveis:

As probabilidades objetivas são obtidas experimentalmente a partir da frequência relativa da ocorrência de um resultado em muitos testes da variável aleatória. Se o processo aleatório for repetido N vezes, e o evento A ocorrer NA vezes, então

Por exemplo, uma série de N = 100, 200, 300 lances de um dado pode resultar em N1 = 19, 30, 48 ocorrências do valor 1.

Os valores 0.19, 0.15, 0.16 fornecem uma estimativa cada vez mais confiável da probabilidade pdado({1}).

36 Probability

From a practical point of view, we would like to know how to assignprobability values to various outcomes. There are two possible approaches:

1. Objective probabilities are obtained experimentally from the relative frequency of the

occurrence of an outcome in many tests of the random variable. If the random process is

repeated N times, and the event A occurs NA times, then

p!A" = limN→"

NA

N#

For example, a series of N = 100$ 200$ 300 throws of a dice may result in N1 = 19$ 30$ 48occurrences of 1. The ratios 0.19, 0.15, 0.16 provide an increasingly more reliable estimateof the probability pdice!%1&".

2. Subjective probabilities provide a theoretical estimate based on the uncertainties relatedto lack of precise knowledge of outcomes. For example, the assessment pdice!%1&" = 1/6is based on the knowledge that there are six possible outcomes to a dice throw, and that inthe absence of any prior reason to believe that the dice is biased, all six are equally likely.All assignments of probability in statistical mechanics are subjectively based. The conse-quences of such subjective assignments of probability have to be checked against measure-ments, and they may need to be modified as more information about the outcomes becomesavailable.

2.2 One random variable

As the properties of a discrete random variable are rather well known, herewe focus on continuous random variables, which are more relevant to ourpurposes. Consider a random variable x, whose outcomes are real numbers,that is, !x = %−" < x < "&.

• The cumulative probability function (CPF) P!x" is the probability of anoutcome with any value less than x, that is, P!x" = prob!E ⊂ '−"$x(". P!x"must be a monotonically increasing function of x, with P!−"" = 0 andP!+"" = 1.

Fig. 2.1 A typicalcumulative probabilityfunction.

P(x)

x0

1

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As probabilidades subjetivas fornecem uma estimativa teórica baseada nas incertezas relacionadas à falta de conhecimento preciso dos resultados.

Por exemplo, a avaliação pdado({1}) = 1/6 baseia-se no conhecimento de que há seis resultados possíveis para um lance de dados, e que na ausência de qualquer razão prévia para acreditar que os dados são tendenciosos, todos os seis são igualmente prováveis.

Todas as atribuições de probabilidade em mecânica estatística são subjetivamente baseadas ➜ e.g. princípio de igual probabilidade a priori.

As consequências de tais atribuições subjetivas de probabilidade têm que ser posteriormente verificadas através de medições, e elas podem precisar ser modificadas à medida que mais informações sobre os resultados se tornam disponíveis.

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Densidade de probabilidade

Aqui nos concentraremos apenas em variáveis aleatórias contínuas, que são as mais relevantes para nossos propósitos.

Consideremos uma variável aleatória x, cujos resultados são números reais, ou seja,

!x ≡ { −∞< x < +∞}

Exemplo: x pode ser uma das componentes da coordenada ou do momento de uma molécula imersa em um gás.

A função de probabilidade cumulativa (cumulative probability function, CPF) P(x) é a probabilidade de obter um resultado com qualquer valor menor que x, ou seja, P(x) = prob(X ∈ (-∞, x]).

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36 Probability

From a practical point of view, we would like to know how to assignprobability values to various outcomes. There are two possible approaches:

1. Objective probabilities are obtained experimentally from the relative frequency of the

occurrence of an outcome in many tests of the random variable. If the random process is

repeated N times, and the event A occurs NA times, then

p!A" = limN→"

NA

N#

For example, a series of N = 100$ 200$ 300 throws of a dice may result in N1 = 19$ 30$ 48occurrences of 1. The ratios 0.19, 0.15, 0.16 provide an increasingly more reliable estimateof the probability pdice!%1&".

2. Subjective probabilities provide a theoretical estimate based on the uncertainties relatedto lack of precise knowledge of outcomes. For example, the assessment pdice!%1&" = 1/6is based on the knowledge that there are six possible outcomes to a dice throw, and that inthe absence of any prior reason to believe that the dice is biased, all six are equally likely.All assignments of probability in statistical mechanics are subjectively based. The conse-quences of such subjective assignments of probability have to be checked against measure-ments, and they may need to be modified as more information about the outcomes becomesavailable.

2.2 One random variable

As the properties of a discrete random variable are rather well known, herewe focus on continuous random variables, which are more relevant to ourpurposes. Consider a random variable x, whose outcomes are real numbers,that is, !x = %−" < x < "&.

• The cumulative probability function (CPF) P!x" is the probability of anoutcome with any value less than x, that is, P!x" = prob!E ⊂ '−"$x(". P!x"must be a monotonically increasing function of x, with P!−"" = 0 andP!+"" = 1.

Fig. 2.1 A typicalcumulative probabilityfunction.

P(x)

x0

1

Esboço de uma CPF típica.!9

Propriedades da CPF:

- P(x) deve ser uma função monotonamente crescente de x: Se x2 > x1, então P(x2) > P(x1)

- P(-∞) = 0 - P(+∞) = 1.

A função de densidade de probabilidade (probability density function, PDF) é definida por

p(x) ≡ dP(x)/dx.

A densidade de probabilidade é positiva e normalizada de forma tal que

Note que uma vez que p(x) é uma densidade de probabilidade, ela tem dimensões de x-1 e muda seu valor se as unidades que medem x forem modificadas.

2.2 One random variable 37

x0

p(x) =d Pd x

Fig. 2.2 A typicalprobability densityfunction.

• The probability density function (PDF) is defined by p!x" ≡ dP!x"/dx.Hence, p!x"dx = prob!E ∈ #x$x+dx%". As a probability density, it is positive,and normalized such that

prob!!" =! #

−#dx p!x" = 1& (2.1)

Note that since p!x" is a probability density, it has dimensions of #x%−1, andchanges its value if the units measuring x are modified. Unlike P!x", the PDFhas no upper bound, that is, 0 < p!x" < #, and may contain divergences aslong as they are integrable.

• The expectation value of any function, F!x", of the random variable is

%F!x"& =! #

−#dx p!x"F!x"& (2.2)

x0

F

d F

d xx1 x2 x3

Fig. 2.3 Obtaining thePDF for the functionF !x".

The function F!x" is itself a random variable, with an associated PDF ofpF !f"df = prob!F!x" ∈ #f$f +df%". There may be multiple solutions xi to theequation F!x" = f , and

pF !f"df ="

i

p!xi"dxi $ =⇒ pF !f" ="

i

p!xi"

####dx

dF

####x=xi

& (2.3)

The factors of (dx/dF ( are the Jacobians associated with the change of variablesfrom x to F . For example, consider p!x" = ' exp!−'(x("/2, and the function

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2.2 One random variable 37

x0

p(x) =d Pd x

Fig. 2.2 A typicalprobability densityfunction.

• The probability density function (PDF) is defined by p!x" ≡ dP!x"/dx.Hence, p!x"dx = prob!E ∈ #x$x+dx%". As a probability density, it is positive,and normalized such that

prob!!" =! #

−#dx p!x" = 1& (2.1)

Note that since p!x" is a probability density, it has dimensions of #x%−1, andchanges its value if the units measuring x are modified. Unlike P!x", the PDFhas no upper bound, that is, 0 < p!x" < #, and may contain divergences aslong as they are integrable.

• The expectation value of any function, F!x", of the random variable is

%F!x"& =! #

−#dx p!x"F!x"& (2.2)

x0

F

d F

d xx1 x2 x3

Fig. 2.3 Obtaining thePDF for the functionF !x".

The function F!x" is itself a random variable, with an associated PDF ofpF !f"df = prob!F!x" ∈ #f$f +df%". There may be multiple solutions xi to theequation F!x" = f , and

pF !f"df ="

i

p!xi"dxi $ =⇒ pF !f" ="

i

p!xi"

####dx

dF

####x=xi

& (2.3)

The factors of (dx/dF ( are the Jacobians associated with the change of variablesfrom x to F . For example, consider p!x" = ' exp!−'(x("/2, and the function

Esboço de uma PDF típica. !11

Propriedades da PDF:

- por ser a derivada da CPF, a PDF deve tender a zero nos limites x→-∞ e x→+∞.

- Ao contrário de P(x), a PDF não tem limite superior, ou seja, 0 <p(x) < ∞, e pode conter divergências, desde que sejam integráveis.

Interpretação da PDF:

Vejamos o significado de p(x)dx:

Pela definição p(x) ≡ dP(x)/dx, temos: dP(x) = p(x) dx

Mas, dP(x) = P(x+dx) - P(x), portanto:

p(x) dx = P(x+dx) - P(x) = prob(X ∈ [x, x + dx]).

Assim, p(x) dx indica probabilidade de que a variável aleatória X adote um valor entre x e x+dx.

Em outras palavras, p(x) é a probabilidade por "unidade da grandeza x” de encontrar X em torno do valor x.

→por isso, em geral depende das unidades de x !12

O valor esperado ou valor médio da variável aleatória X, é:

Qualquer função F(X) de uma variável aleatória X é também uma variável aleatória. Chamamos F(X) de função aleatória. Seu valor esperado é definido de maneira análoga à expressão anterior:

No caso de uma variável aleatória discreta temos:

⟨X⟩ = ∑i xi p(xi). ⟨F(X)⟩ = ∑i F(xi) p(xi).

hXi =Z +1

�1x p(x) dx

hF (X)i =Z +1

�1F (x) p(x) dx

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Valor esperado ou valor médio

Momentos de uma PDF

O n-ésimo momento da densidade de probabilidade p(x) é definido como:

Isto é, o momento de ordem n da PDF de uma variável aleatória X é o valor esperado da função Xn.

Os momentos ⟨Xn⟩ (n = 0, 1, 2, ...) servem para revelar as propriedades de X.

Também é útil definir os momentos em torno da média ⟨(X - ⟨X⟩)n⟩, também denominados momentos centrados.

µn = hXni =Z +1

�1xn p(x) dx

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Analisemos os primeiros momentos centrados:

O momento centrado de ordem zero é:

⟨(x - ⟨X⟩)0⟩ = ⟨1⟩ = ∫ 1 . p(x) dx = 1.

Simplesmente recuperamos o fato de que as probabilidades são normalizadas.

O momento centrado de primeira ordem é:

⟨(x - ⟨X⟩)1⟩ = ∫ (x - ⟨X⟩)1.p(x)dx = ∫ x p(x)dx − ⟨X⟩ ∫ p(x)dx = ⟨X⟩ − ⟨X⟩.1 = 0

onde foi usada a propriedade de normalização e a definição de média.

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Momentos centrados de ordem superior (com n ≥ 2) descrevem outras propriedades de X.

O segundo momento centrado é denominado variância de X:

var{X} = ⟨(X − ⟨X⟩)2⟩.

Usando a linearidade do operador ⟨ ⟩, temos:

8 EXPECTED VALUES

=!

ixi P(xi ) − ⟨X⟩

!

iP(xi )

= ⟨X⟩ − ⟨X⟩⟨1⟩

= 0 (2.1.4)

follows from normalization (2.1.3) and the definition of the mean (2.1.1).Higher order moments (with n ≥ 2 ) describe other properties of X . Forinstance, the second moment about the mean or the variance of X , denoted byvar{X} and defined by

var{X} = ⟨(X − ⟨X⟩)2⟩, (2.1.5)

quantifies the variability, or mean squared deviation, of X from its mean ⟨X⟩.The linearity of the expected value operator ⟨⟩ (see section 2.2) ensures that(2.1.5) reduces to

var{X} = ⟨X2 − 2X⟨X⟩ + ⟨X⟩2⟩

= ⟨X2⟩ − 2⟨X⟩2 + ⟨X⟩2

= ⟨X2⟩ − ⟨X⟩2. (2.1.6)

The mean and variance are sometimes denoted by the Greek letters µ and σ 2,respectively, and

√σ 2 = σ is called the standard deviation of X . The third

moment about the mean enters into the definition of skewness,

skewness{X} = ⟨(X − µ)3⟩σ 3

, (2.1.7)

and the fourth moment into the kurtosis,

kurtosis{X} = ⟨(X − µ)4⟩σ 4

. (2.1.8)

The skewness and kurtosis are dimensionless shape parameters. The formerquantifies the asymmetry of X around its mean, while the latter is a measure ofthe degree to which a given variance σ 2 is accompanied by realizations of Xclose to (relatively small kurtosis) and far from (large kurtosis) µ ± σ . Highlypeaked and long-tailed probability functions have large kurtosis; broad, squatones have small kurtosis. See Problem 2.1, Dice Parameters, for practice incalculating parameters.

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A média e a variância são denominadas às vezes com as letras " e #2:

⟨X⟩ ≡ " var{X} ≡ #2

A raiz quadrada da variância é denominada desvio padrão:

# = ( ⟨(X − ⟨X⟩)2⟩ )1/2

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O terceiro momento centrado entra na definição da skewness (em português, obliquidade ou assimetria):

e o quarto momento, entra na definição da kurtosis:

A skewness e a kurtosis são parâmetros adimensionais que medem a forma das caudas da distribuição.

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skewness{X} =h(X � hXi)3i

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kurtosis{X} =h(X � hXi)4i

�4<latexit sha1_base64="40/Qq4dxre4dEWMlaeRGAy+QFQQ=">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</latexit><latexit sha1_base64="40/Qq4dxre4dEWMlaeRGAy+QFQQ=">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</latexit><latexit sha1_base64="40/Qq4dxre4dEWMlaeRGAy+QFQQ=">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</latexit><latexit sha1_base64="40/Qq4dxre4dEWMlaeRGAy+QFQQ=">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</latexit>

A skewness quantifica a assimetria de X em torno de sua média:

•Se skewness{X}>0, a distribuição tem uma cauda direita mais longa. •Se skewness{X}<0, a distribuição tem uma cauda esquerda mais longa. •Se skewness{X}=0, a distribuição é aproximadamente simétrica (na terceira potência do desvio em relação à média).

Capítulo 2. Turbulência 20

ção em relação a média. Utilizando a eq. (2.2.6) para m = 3 teremos que

S = hu3i =

hu03i

�3u

=1

N

NX

n=1

un � hui

�u

�3. (2.2.8)

Assim, para uma função simétrica (e.g. gaussiana) o valor de S será nulo. Casoa distribuição possua S > 0, a cauda da função estará deslocada para a direita do va-lor médio, enquanto S < 0 à esquerda. Na fig. 2.3(a) podemos ver uma representaçãoesquemática de distribuições com obliquidades opostas.

(a) Obliquidade (b) Curtose

Figura 2.3: Diagrama Esquemático da Obliquidade e da Curtose. Os valores S = 0 e K = 3 (linhaspontilhadas) representam uma distribuição gaussiana.

E por último, o momento central de quarta ordem

K = hu4i =

hu04i

�4u

=1

N

NX

n=1

un � hui

�u

�4, (2.2.9)

chamado de curtose (em inglês kurtosis), indica o quão achatada ou pontiaguda é adistribuição. Alguns autores costumam também utilizar-se da planura definida comoF = K � 3, que relaciona a distribuição com uma função gaussiana. Chamamos demesocúrtica uma distribuição que possua K = 3 (ou F = 0). Para K > 3 (ou F > 0)a distribuição apresentará um formato mais afunilado e concentrado que a distribuiçãonormal, enquanto K < 3 (ou F > 0) indicará um perfil mais plano. Essas distribuiçõespodem ser classificadas, respectivamente, como leptocúrticas e platicúrticas.

!19

A kurtosis é uma medida de forma que caracteriza o achatamento da curva da função de distribuição de probabilidade.

• Se K= 3, a distribuição tem o mesmo achatamento que a distribuição normal.

• Se K>3, então a distribuição em questão é mais alta (afunilada) e concentrada que a distribuição normal. Neste caso, é relativamente fácil obter valores que não se aproximam da média a vários múltiplos do desvio padrão.

• Se K<3, a função de distribuição é mais "achatada" que a distribuição normal.

Capítulo 2. Turbulência 20

ção em relação a média. Utilizando a eq. (2.2.6) para m = 3 teremos que

S = hu3i =

hu03i

�3u

=1

N

NX

n=1

un � hui

�u

�3. (2.2.8)

Assim, para uma função simétrica (e.g. gaussiana) o valor de S será nulo. Casoa distribuição possua S > 0, a cauda da função estará deslocada para a direita do va-lor médio, enquanto S < 0 à esquerda. Na fig. 2.3(a) podemos ver uma representaçãoesquemática de distribuições com obliquidades opostas.

(a) Obliquidade (b) Curtose

Figura 2.3: Diagrama Esquemático da Obliquidade e da Curtose. Os valores S = 0 e K = 3 (linhaspontilhadas) representam uma distribuição gaussiana.

E por último, o momento central de quarta ordem

K = hu4i =

hu04i

�4u

=1

N

NX

n=1

un � hui

�u

�4, (2.2.9)

chamado de curtose (em inglês kurtosis), indica o quão achatada ou pontiaguda é adistribuição. Alguns autores costumam também utilizar-se da planura definida comoF = K � 3, que relaciona a distribuição com uma função gaussiana. Chamamos demesocúrtica uma distribuição que possua K = 3 (ou F = 0). Para K > 3 (ou F > 0)a distribuição apresentará um formato mais afunilado e concentrado que a distribuiçãonormal, enquanto K < 3 (ou F > 0) indicará um perfil mais plano. Essas distribuiçõespodem ser classificadas, respectivamente, como leptocúrticas e platicúrticas.

!20

A função característica é simplesmente a transformada de Fourier da PDF (função de distribuição de probabilidade):

De acordo com a definição do valor médio, vemos que a função característica é o valor esperado da função exp(–ikx):

A PDF pode ser recuperada a partir da função característica através da transformada de Fourier inversa:

Função característica

p(k) =

Z +1

�1e�ikx p(x) dx

p(k) = he�ikxi

p(x) =1

2⇡

Z +1

�1e+ikx p(k) dk

!21

Teorema: A função característica pode ser escrita em série de potências de k, com coeficientes proporcionais aos momentos da PDF.

Demonstração: Expandindo a exponencial em série de potências temos:

p(k) = he�ikxi = h1X

n=0

(�ikx)n

n!i =

1X

n=0

(�ik)n

n!hxni

!22

p(k) =1X

n=0

(�ik)n

n!hxni

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Definimos a função geratriz dos cumulantes como o logaritmo da função característica:

Os cumulantes de uma PDF são um conjunto de grandezas definidas através da expressão:

Eq. (*)

onde ⟨xn⟩c é o cumulante de ordem n.

Isto é, o logaritmo da função característica pode ser escrito em série de potências de k, com coeficientes proporcionais aos cumulantes da PDF.

Assim como os momentos da PDF, os cumulantes fornecem uma descrição das propriedades de uma PDF.

Cumulantes

ln p(k) =1X

n=1

(�ik)n

n!hxnic

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!23

H(k) = ln p(k)<latexit sha1_base64="w1wVU5drB8Ttm07PT1Ho9yMN4Lg=">AAAB/3icbVBNS8NAEN34WetX1IMHL4tFqJeSiqAehKKXHisYW2hC2Ww37dLNJuxOhBJy8a948aDi1b/hzX/jts1BWx8MPN6bYWZekAiuwXG+raXlldW19dJGeXNre2fX3tt/0HGqKHNpLGLVCYhmgkvmAgfBOoliJAoEawej24nffmRK81jewzhhfkQGkoecEjBSzz5sVken+Bp7QmIPuOizLMmN1LMrTs2ZAi+SekEqqECrZ395/ZimEZNABdG6W3cS8DOigFPB8rKXapYQOiID1jVUkohpP5s+kOMTo/RxGCtTEvBU/T2RkUjrcRSYzojAUM97E/E/r5tCeOlnXCYpMElni8JUYIjxJA3c54pREGNDCFXc3IrpkChCwWRWNiHU519eJO5Z7arm3J1XGjdFGiV0hI5RFdXRBWqgJmohF1GUo2f0it6sJ+vFerc+Zq1LVjFzgP7A+vwBw96UzQ==</latexit><latexit sha1_base64="w1wVU5drB8Ttm07PT1Ho9yMN4Lg=">AAAB/3icbVBNS8NAEN34WetX1IMHL4tFqJeSiqAehKKXHisYW2hC2Ww37dLNJuxOhBJy8a948aDi1b/hzX/jts1BWx8MPN6bYWZekAiuwXG+raXlldW19dJGeXNre2fX3tt/0HGqKHNpLGLVCYhmgkvmAgfBOoliJAoEawej24nffmRK81jewzhhfkQGkoecEjBSzz5sVken+Bp7QmIPuOizLMmN1LMrTs2ZAi+SekEqqECrZ395/ZimEZNABdG6W3cS8DOigFPB8rKXapYQOiID1jVUkohpP5s+kOMTo/RxGCtTEvBU/T2RkUjrcRSYzojAUM97E/E/r5tCeOlnXCYpMElni8JUYIjxJA3c54pREGNDCFXc3IrpkChCwWRWNiHU519eJO5Z7arm3J1XGjdFGiV0hI5RFdXRBWqgJmohF1GUo2f0it6sJ+vFerc+Zq1LVjFzgP7A+vwBw96UzQ==</latexit><latexit sha1_base64="w1wVU5drB8Ttm07PT1Ho9yMN4Lg=">AAAB/3icbVBNS8NAEN34WetX1IMHL4tFqJeSiqAehKKXHisYW2hC2Ww37dLNJuxOhBJy8a948aDi1b/hzX/jts1BWx8MPN6bYWZekAiuwXG+raXlldW19dJGeXNre2fX3tt/0HGqKHNpLGLVCYhmgkvmAgfBOoliJAoEawej24nffmRK81jewzhhfkQGkoecEjBSzz5sVken+Bp7QmIPuOizLMmN1LMrTs2ZAi+SekEqqECrZ395/ZimEZNABdG6W3cS8DOigFPB8rKXapYQOiID1jVUkohpP5s+kOMTo/RxGCtTEvBU/T2RkUjrcRSYzojAUM97E/E/r5tCeOlnXCYpMElni8JUYIjxJA3c54pREGNDCFXc3IrpkChCwWRWNiHU519eJO5Z7arm3J1XGjdFGiV0hI5RFdXRBWqgJmohF1GUo2f0it6sJ+vFerc+Zq1LVjFzgP7A+vwBw96UzQ==</latexit>

Por definição temos:

onde ⟨xn⟩c é o cumulante de ordem n.

Por outro lado, sabemos que a função característica pode ser escrita como:

Tomando logaritmo na expressão anterior:

Relação entre momentos e cumulantes

ln p(k) =1X

n=1

(�ik)n

n!hxnic

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p(k) =1X

n=0

(�ik)n

n!hxni

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ln p(k) = ln1X

n=0

(�ik)n

n!hxni = ln

1 +

1X

n=1

(�ik)n

n!hxni

!

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!24

Eq. (*)

Agora expandimos em série o logaritmo do lado direito, usando:

Obtemos:

Comparando última equação com a Eq. (*) do slide anterior, temos:

Esta expressão relaciona os cumulantes com os momentos da distribuição.

ln p(k) = ln

"1 +

1X

n=1

(�ik)n

n!hxni

!#=

1X

`=1

(�1)`+1

`

1X

n=1

(�ik)n

n!hxni

!`

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ln(1 + ✏) =1X

`=1

(�1)`+1

`✏`

<latexit sha1_base64="Em2DiIEQJLy3v2FPqcrDIEwFygI=">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</latexit><latexit sha1_base64="Em2DiIEQJLy3v2FPqcrDIEwFygI=">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</latexit><latexit sha1_base64="Em2DiIEQJLy3v2FPqcrDIEwFygI=">AAACQHicbZBLSwMxFIUz9VXra9Slm2ARWoplRgR1USi6calgtdBpSya904ZmMkOSEcowf82N/8CdezcuVNy6Mn0gvi4EDufce5N8fsyZ0o7zaOXm5hcWl/LLhZXVtfUNe3PrWkWJpNCgEY9k0ycKOBPQ0ExzaMYSSOhzuPGHZ+P85hakYpG40qMY2iHpCxYwSrSxunbT46Lk4gr2IFaMR6KMawVPJWE39YBzXMNu1kk9JgI9yrAXSELT0r5b7kzjiptlE5V9LZgmWdcuOlVnUvivcGeiiGZ10bUfvF5EkxCEppwo1XKdWLdTIjWjHLKClyiICR2SPrSMFCQE1U4nBDK8Z5weDiJpjtB44n6fSEmo1Cj0TWdI9ED9zsbmf1kr0cFxO2UiTjQIOr0oSDjWER7jxD0mgWo+MoJQycxbMR0QQ0kb6AUDwf395b+icVA9qTqXh8X66YxGHu2gXVRCLjpCdXSOLlADUXSHntALerXurWfrzXqftuas2cw2+lHWxyfwHK8Z</latexit>

!25

1X

m=1

(�ik)m

m!hxmic =

1X

`=1

(�1)`+1

`

1X

n=1

(�ik)n

n!hxni

!`

<latexit sha1_base64="2fyLcUBrqwVpOzX4HC55QmLKp2A=">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</latexit><latexit sha1_base64="2fyLcUBrqwVpOzX4HC55QmLKp2A=">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</latexit><latexit sha1_base64="2fyLcUBrqwVpOzX4HC55QmLKp2A=">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</latexit>

Eq. (**)

Fixando um valor de m, podemos obter uma expressão para o cumulante de ordem m, em função de momentos da distribuição.

Para isso, devemos relacionar potências iguais de (ik) em ambos lados da equação anterior. Isto é, o coeficiente que multiplica (ik)m no lado esquerdo deve ser igual à soma dos coeficientes que multiplicam a (ik)nl no lado direito, para m=nl.

Comecemos com os termos proporcionais a k:

!26

(�ik)

1!hxic

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(�1)2

1

(�ik)1

1!hxi

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Do lado esquerdo da Eq. (**) do slide anterior o termo proporcional a k1 é:

Do lado direito da Eq. (**) o termo proporcional a k1 é obtido com n=1 e l=1:

Portanto, o primeiro cumulante é sempre igual à média:

hxic = hxi<latexit sha1_base64="QQ/L1iOMcMOBVafbgPqLZBvCfoM=">AAACFXicbZDNSsNAFIUn/tb6F3XpZrAIbiypCOpCKLpxWcHYQhPKZHrTDp1MwsxELKFP4cZXceNCxa3gzrdxmmahbS9c+DjnXmbuCRLOlHacH2thcWl5ZbW0Vl7f2Nzatnd271WcSgoujXksWwFRwJkAVzPNoZVIIFHAoRkMrsd+8wGkYrG408ME/Ij0BAsZJdpIHfsYe5yIHgf8iD2ZU4dijC9Nz7HsilN18sKzUCuggopqdOxvrxvTNAKhKSdKtWtOov2MSM0oh1HZSxUkhA5ID9oGBYlA+Vl+1ggfGqWLw1iaFhrn6t+NjERKDaPATEZE99W0Nxbnee1Uh+d+xkSSahB08lCYcqxjPM4Id5kEqvnQAKGSmb9i2ieSUG2SLJsQatMnz4J7Ur2oOrenlfpVkUYJ7aMDdIRq6AzV0Q1qIBdR9IRe0Bt6t56tV+vD+pyMLljFzh76V9bXL/lbnPc=</latexit><latexit sha1_base64="QQ/L1iOMcMOBVafbgPqLZBvCfoM=">AAACFXicbZDNSsNAFIUn/tb6F3XpZrAIbiypCOpCKLpxWcHYQhPKZHrTDp1MwsxELKFP4cZXceNCxa3gzrdxmmahbS9c+DjnXmbuCRLOlHacH2thcWl5ZbW0Vl7f2Nzatnd271WcSgoujXksWwFRwJkAVzPNoZVIIFHAoRkMrsd+8wGkYrG408ME/Ij0BAsZJdpIHfsYe5yIHgf8iD2ZU4dijC9Nz7HsilN18sKzUCuggopqdOxvrxvTNAKhKSdKtWtOov2MSM0oh1HZSxUkhA5ID9oGBYlA+Vl+1ggfGqWLw1iaFhrn6t+NjERKDaPATEZE99W0Nxbnee1Uh+d+xkSSahB08lCYcqxjPM4Id5kEqvnQAKGSmb9i2ieSUG2SLJsQatMnz4J7Ur2oOrenlfpVkUYJ7aMDdIRq6AzV0Q1qIBdR9IRe0Bt6t56tV+vD+pyMLljFzh76V9bXL/lbnPc=</latexit><latexit sha1_base64="QQ/L1iOMcMOBVafbgPqLZBvCfoM=">AAACFXicbZDNSsNAFIUn/tb6F3XpZrAIbiypCOpCKLpxWcHYQhPKZHrTDp1MwsxELKFP4cZXceNCxa3gzrdxmmahbS9c+DjnXmbuCRLOlHacH2thcWl5ZbW0Vl7f2Nzatnd271WcSgoujXksWwFRwJkAVzPNoZVIIFHAoRkMrsd+8wGkYrG408ME/Ij0BAsZJdpIHfsYe5yIHgf8iD2ZU4dijC9Nz7HsilN18sKzUCuggopqdOxvrxvTNAKhKSdKtWtOov2MSM0oh1HZSxUkhA5ID9oGBYlA+Vl+1ggfGqWLw1iaFhrn6t+NjERKDaPATEZE99W0Nxbnee1Uh+d+xkSSahB08lCYcqxjPM4Id5kEqvnQAKGSmb9i2ieSUG2SLJsQatMnz4J7Ur2oOrenlfpVkUYJ7aMDdIRq6AzV0Q1qIBdR9IRe0Bt6t56tV+vD+pyMLljFzh76V9bXL/lbnPc=</latexit>

Do lado direito da Eq. (**) temos: - para l=1 o único termo proporcional a k2 é obtido com n=2. - para l=2 o único termo proporcional a k2 é obtido com n=1. - os outros termos são de ordem superior.

Igualando as expressões acima, vemos que o segundo cumulante é sempre igual à variância:

Consideremos os termos proporcionais a k2:

!27

Do lado esquerdo da Eq. (**) usamos m=2: (�ik)2

2!hx2ic

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(�1)2

1

(�ik)2

2!hx2i

�1+

(�1)3

2

(�ik)1

1!hxi

�2

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hx2ic = hx2i � hxi2 = var{x}<latexit sha1_base64="5Hs7JhynLOarXbo0UXA2q6lcuNg=">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</latexit><latexit sha1_base64="5Hs7JhynLOarXbo0UXA2q6lcuNg=">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</latexit><latexit sha1_base64="5Hs7JhynLOarXbo0UXA2q6lcuNg=">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</latexit>

Dessa forma obtemos que os primeiros quatro cumulantes são a média, a variância, a skewness, e a kurtosis:

!28

2.2 One random variable 39

• The cumulant generating function is the logarithm of the characteristic func-tion. Its expansion generates the cumulants of the distribution defined through

ln p!k" =!!

n=1

!−ik"n

n! #xn$c # (2.9)

Relations between moments and cumulants can be obtained by expandingthe logarithm of p!k" in Eq. (2.7 ), and using

ln!1+ $" =!!

n=1

!−1"n+1 $n

n# (2.10)

The first four cumulants are called the mean, variance, skewness, and curtosis(or kurtosis) of the distribution, respectively, and are obtained from themoments as

#x$c = #x$ %

"x2#

c="x2#−#x$2 %

"x3 #

c="x3 #− 3

"x2# #x$+2 #x$3 %

"x4 #

c="x4 #− 4

"x3 # #x$− 3

"x2#2 +12

"x2# #x$2 − 6 #x$4 #

(2.11)

The cumulants provide a useful and compact way of describing a PDF.

An important theorem allows easy computation of moments in terms of thecumulants: represent the nth cumulant graphically as a connected cluster of npoints. The mth moment is then obtained by summing all possible subdivisionsof m points into groupings of smaller (connected or disconnected) clusters. Thecontribution of each subdivision to the sum is the product of the connectedcumulants that it represents. Using this result, the first four moments arecomputed graphically.

< x

> =

< x > =

< x > =

+

+

< x > =

+

+ 3

+ 4 + 3 + 64

3

2

Fig. 2.5 Graphicalcomputation of the firstfour moments.

The corresponding algebraic expressions are

#x$ = #x$c %

"x2#=

"x2#

c+ #x$2

c %

"x3 #=

"x3 #

c+ 3

"x2#

c#x$c + #x$3

c %

"x4 #=

"x4 #

c+ 4

"x3 #

c#x$c + 3

"x2#2

c+ 6

"x2#

c#x$2

c + #x$4c #

(2.12)

Usando um procedimento análogo ao anterior (que não demonstraremos) é possível encontrar expressões para os momentos em função dos cumulantes:

!29

2.2 One random variable 39

• The cumulant generating function is the logarithm of the characteristic func-tion. Its expansion generates the cumulants of the distribution defined through

ln p!k" =!!

n=1

!−ik"n

n! #xn$c # (2.9)

Relations between moments and cumulants can be obtained by expandingthe logarithm of p!k" in Eq. (2.7 ), and using

ln!1+ $" =!!

n=1

!−1"n+1 $n

n# (2.10)

The first four cumulants are called the mean, variance, skewness, and curtosis(or kurtosis) of the distribution, respectively, and are obtained from themoments as

#x$c = #x$ %

"x2#

c="x2#−#x$2 %

"x3 #

c="x3 #− 3

"x2# #x$+2 #x$3 %

"x4 #

c="x4 #− 4

"x3 # #x$− 3

"x2#2 +12

"x2# #x$2 − 6 #x$4 #

(2.11)

The cumulants provide a useful and compact way of describing a PDF.

An important theorem allows easy computation of moments in terms of thecumulants: represent the nth cumulant graphically as a connected cluster of npoints. The mth moment is then obtained by summing all possible subdivisionsof m points into groupings of smaller (connected or disconnected) clusters. Thecontribution of each subdivision to the sum is the product of the connectedcumulants that it represents. Using this result, the first four moments arecomputed graphically.

< x

> =

< x > =

< x > =

+

+

< x > =

+

+ 3

+ 4 + 3 + 64

3

2

Fig. 2.5 Graphicalcomputation of the firstfour moments.

The corresponding algebraic expressions are

#x$ = #x$c %

"x2#=

"x2#

c+ #x$2

c %

"x3 #=

"x3 #

c+ 3

"x2#

c#x$c + #x$3

c %

"x4 #=

"x4 #

c+ 4

"x3 #

c#x$c + 3

"x2#2

c+ 6

"x2#

c#x$2

c + #x$4c #

(2.12)

A distribuição normal (ou Gaussiana) é definida como:

A função característica também tem forma Gaussiana:

Os cumulantes da distribuição podem ser identificados tomando o logaritmo da expressão anterior:

Comparando com a Eq. (*) do slide 24, vemos que o primeiro cumulante é o fator que acompanha a (–ik) e o segundo cumulante é o fator que acompanha a (–k2)/2!. Logo:

A distribuição normal

40 Probability

This theorem, which is the starting point for various diagrammatic computationsin statistical mechanics and field theory, is easily proved by equating theexpressions in Eqs (2.7) and (2.9 ) for p!k":

!!

m=0

!−ik"m

m! #xm$ = exp

"!!

n=1

!−ik"n

n! #xn$c

#

=$

n

!

pn

%!−ik"npn

pn!

& #xn$c

n!

'pn(

# (2.13)

Matching the powers of !−ik"m on the two sides of the above expression leads to

#xm$ =′!

$pn%

m!$

n

1pn!!n!"pn

#xn$pnc # (2.14)

The sum is restricted such that)

npn = m, and leads to the graphical interpre-tation given above, as the numerical factor is simply the number of ways ofbreaking m points into $pn% clusters of n points.

2.3 Some important probability distributions

The properties of three commonly encountered probability distributions areexamined in this section.

(1) The normal (Gaussian) distribution describes a continuous real randomvariable x, with

p!x" = 1√2&'2

exp%− !x−("2

2'2

(# (2.15)

The corresponding characteristic function also has a Gaussian form,

p!k" =* !

−!dx

1√2&'2

exp%− !x−("2

2'2− ikx

(= exp

%−ik(− k2'2

2

(# (2.16)

Cumulants of the distribution can be identified from ln p!k" = −ik(−k2'2/2,using Eq. (2.9 ), as

#x$c = ()+x2,

c= '2)

+x3,

c=+x4,

c= · · · = 0 # (2.17)

The normal distribution is thus completely specified by its two first cumu-lants. This makes the computation of moments using the cluster expansion(Eqs. (2.12)) particularly simple, and

#x$ =()

+x2,='2 +(2)

+x3,=3'2(+(3)

+x4,=3'4 +6'2(2 +(4)

· · ·

(2.18)

40 Probability

This theorem, which is the starting point for various diagrammatic computationsin statistical mechanics and field theory, is easily proved by equating theexpressions in Eqs (2.7) and (2.9 ) for p!k":

!!

m=0

!−ik"m

m! #xm$ = exp

"!!

n=1

!−ik"n

n! #xn$c

#

=$

n

!

pn

%!−ik"npn

pn!

& #xn$c

n!

'pn(

# (2.13)

Matching the powers of !−ik"m on the two sides of the above expression leads to

#xm$ =′!

$pn%

m!$

n

1pn!!n!"pn

#xn$pnc # (2.14)

The sum is restricted such that)

npn = m, and leads to the graphical interpre-tation given above, as the numerical factor is simply the number of ways ofbreaking m points into $pn% clusters of n points.

2.3 Some important probability distributions

The properties of three commonly encountered probability distributions areexamined in this section.

(1) The normal (Gaussian) distribution describes a continuous real randomvariable x, with

p!x" = 1√2&'2

exp%− !x−("2

2'2

(# (2.15)

The corresponding characteristic function also has a Gaussian form,

p!k" =* !

−!dx

1√2&'2

exp%− !x−("2

2'2− ikx

(= exp

%−ik(− k2'2

2

(# (2.16)

Cumulants of the distribution can be identified from ln p!k" = −ik(−k2'2/2,using Eq. (2.9 ), as

#x$c = ()+x2,

c= '2)

+x3,

c=+x4,

c= · · · = 0 # (2.17)

The normal distribution is thus completely specified by its two first cumu-lants. This makes the computation of moments using the cluster expansion(Eqs. (2.12)) particularly simple, and

#x$ =()

+x2,='2 +(2)

+x3,=3'2(+(3)

+x4,=3'4 +6'2(2 +(4)

· · ·

(2.18)

!30

40 Probability

This theorem, which is the starting point for various diagrammatic computationsin statistical mechanics and field theory, is easily proved by equating theexpressions in Eqs (2.7) and (2.9 ) for p!k":

!!

m=0

!−ik"m

m! #xm$ = exp

"!!

n=1

!−ik"n

n! #xn$c

#

=$

n

!

pn

%!−ik"npn

pn!

& #xn$c

n!

'pn(

# (2.13)

Matching the powers of !−ik"m on the two sides of the above expression leads to

#xm$ =′!

$pn%

m!$

n

1pn!!n!"pn

#xn$pnc # (2.14)

The sum is restricted such that)

npn = m, and leads to the graphical interpre-tation given above, as the numerical factor is simply the number of ways ofbreaking m points into $pn% clusters of n points.

2.3 Some important probability distributions

The properties of three commonly encountered probability distributions areexamined in this section.

(1) The normal (Gaussian) distribution describes a continuous real randomvariable x, with

p!x" = 1√2&'2

exp%− !x−("2

2'2

(# (2.15)

The corresponding characteristic function also has a Gaussian form,

p!k" =* !

−!dx

1√2&'2

exp%− !x−("2

2'2− ikx

(= exp

%−ik(− k2'2

2

(# (2.16)

Cumulants of the distribution can be identified from ln p!k" = −ik(−k2'2/2,using Eq. (2.9 ), as

#x$c = ()+x2,

c= '2)

+x3,

c=+x4,

c= · · · = 0 # (2.17)

The normal distribution is thus completely specified by its two first cumu-lants. This makes the computation of moments using the cluster expansion(Eqs. (2.12)) particularly simple, and

#x$ =()

+x2,='2 +(2)

+x3,=3'2(+(3)

+x4,=3'4 +6'2(2 +(4)

· · ·

(2.18)

40 Probability

This theorem, which is the starting point for various diagrammatic computationsin statistical mechanics and field theory, is easily proved by equating theexpressions in Eqs (2.7) and (2.9 ) for p!k":

!!

m=0

!−ik"m

m! #xm$ = exp

"!!

n=1

!−ik"n

n! #xn$c

#

=$

n

!

pn

%!−ik"npn

pn!

& #xn$c

n!

'pn(

# (2.13)

Matching the powers of !−ik"m on the two sides of the above expression leads to

#xm$ =′!

$pn%

m!$

n

1pn!!n!"pn

#xn$pnc # (2.14)

The sum is restricted such that)

npn = m, and leads to the graphical interpre-tation given above, as the numerical factor is simply the number of ways ofbreaking m points into $pn% clusters of n points.

2.3 Some important probability distributions

The properties of three commonly encountered probability distributions areexamined in this section.

(1) The normal (Gaussian) distribution describes a continuous real randomvariable x, with

p!x" = 1√2&'2

exp%− !x−("2

2'2

(# (2.15)

The corresponding characteristic function also has a Gaussian form,

p!k" =* !

−!dx

1√2&'2

exp%− !x−("2

2'2− ikx

(= exp

%−ik(− k2'2

2

(# (2.16)

Cumulants of the distribution can be identified from ln p!k" = −ik(−k2'2/2,using Eq. (2.9 ), as

#x$c = ()+x2,

c= '2)

+x3,

c=+x4,

c= · · · = 0 # (2.17)

The normal distribution is thus completely specified by its two first cumu-lants. This makes the computation of moments using the cluster expansion(Eqs. (2.12)) particularly simple, and

#x$ =()

+x2,='2 +(2)

+x3,=3'2(+(3)

+x4,=3'4 +6'2(2 +(4)

· · ·

(2.18)

Portanto, para a distribuição normal, apenas os dois primeiros cumulantes são não nulos. Aliás, a distribuição normal é a única distribuição que tem todos os cumulantes nulos menos o primeiro e o segundo.

Isto facilita muito o cálculo dos momentos a partir dos cumulantes. Usando as expressões do slide 29, obtemos:

!31

40 Probability

This theorem, which is the starting point for various diagrammatic computationsin statistical mechanics and field theory, is easily proved by equating theexpressions in Eqs (2.7) and (2.9 ) for p!k":

!!

m=0

!−ik"m

m! #xm$ = exp

"!!

n=1

!−ik"n

n! #xn$c

#

=$

n

!

pn

%!−ik"npn

pn!

& #xn$c

n!

'pn(

# (2.13)

Matching the powers of !−ik"m on the two sides of the above expression leads to

#xm$ =′!

$pn%

m!$

n

1pn!!n!"pn

#xn$pnc # (2.14)

The sum is restricted such that)

npn = m, and leads to the graphical interpre-tation given above, as the numerical factor is simply the number of ways ofbreaking m points into $pn% clusters of n points.

2.3 Some important probability distributions

The properties of three commonly encountered probability distributions areexamined in this section.

(1) The normal (Gaussian) distribution describes a continuous real randomvariable x, with

p!x" = 1√2&'2

exp%− !x−("2

2'2

(# (2.15)

The corresponding characteristic function also has a Gaussian form,

p!k" =* !

−!dx

1√2&'2

exp%− !x−("2

2'2− ikx

(= exp

%−ik(− k2'2

2

(# (2.16)

Cumulants of the distribution can be identified from ln p!k" = −ik(−k2'2/2,using Eq. (2.9 ), as

#x$c = ()+x2,

c= '2)

+x3,

c=+x4,

c= · · · = 0 # (2.17)

The normal distribution is thus completely specified by its two first cumu-lants. This makes the computation of moments using the cluster expansion(Eqs. (2.12)) particularly simple, and

#x$ =()

+x2,='2 +(2)

+x3,=3'2(+(3)

+x4,=3'4 +6'2(2 +(4)

· · ·

(2.18)