Formulary for Statistical Data Analysis

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Contents

1 Probability theory
1.1 Combinatorics
1.2 Probability
1.3 Random variables and distributions
1.4 Important distributions
1.4.1 Binomial distribution
1.4.2 Poisson distribution
1.4.3 Uniform distribution
1.4.4 Exponential distribution
1.4.5 Gaussian distribution
1.4.6 Multivariate Gaussian distribution
1.4.7 Chi-square distribution
1.4.8 Cauchy (Breit-Wigner) distribution
1.4.9 Student’s t distribution

Chapter 1
Probability theory

1.1 Combinatorics

Permutations without repetitions:

Pn = n!
(1.1)

Permutations with repetitions:

Pr = ∏-n!- n ki!
(1.2)

Dispositions without repetitions:

n! Dn,k = -------- (n - k)!
(1.3)

Dispositions with repetitions:

Drn,k = nk
(1.4)

Combinations without repetitions:

( ) n ----n!---- Cn,k = k = k!(n - k)!
(1.5)

Combinations with repetitions:

( ) Cr = n + k - 1 n,k k
(1.6)

1.2 Probability

Conditional probability:

P(A ∩ B) P (A |B ) = ---------- p(B )
(1.7)

Probability of intersection for independent events:

P(A ∩ B ) = P (A)P (B )
(1.8)

Bayes’ theorem:

P (A|B ) = P-(B|A-)P(A-) P (B )
(1.9)

Law of total probability:

--P-(B|A-)P(A-)--- P (A |B ) = ∑ P (B |Ai )P (Ai) i
(1.10)

Bayes’ theorem in Baeysian thinking:

P (⃗x|H )π (H ) P (H |⃗x) = ∫----------------- P (⃗x|H )π(H )dH
(1.11)

1.3 Random variables and distributions

Marginal pdf:

∫ ∏ fi(xi) = f(⃗x) dxj j
(1.12)

Conditional pdfs:

f (x,y) f(x,y) f(y|x) = -------f(x |y) = ------- fx(x) fy(y)
(1.13)

Bayes’ theorem for distributions

f(x|y) = f(y|x)fx(x)- fy(y)
(1.14)

Condition for independent variables:

f (x,y) = f (x)f (y) x y
(1.15)

Distribution of a function of a random variable in 1-D:

dx g(a) = f(x(a))|---| da
(1.16)

Distribution of a function of a random variable in N-D:

g(⃗y) = |J|f(⃗x)
(1.17)

Expectation value:

∫ E [x] = xf (x)dx = μx
(1.18)

Variance:

2 2 V [x ] = E [x ] - E [x]
(1.19)

Covariance:

cov[x,y] = E [xy ] - E [x]E[y] = E [(x - μx )(y - μy )]
(1.20)

Correlation coefficient:

cov[xy] ρxy = ------- σxσy
(1.21)

Correlation matrix:

V = cov[x ,x ] ij i j
(1.22)

Variance of a function of random variables:

∑ [ ∂y ∂y ] σ2y ≈ -------- Vij i,j ∂xi ∂xj ⃗x= ⃗μ
(1.23)

Variance of a vector function of random variables:

[ ] ∑ ∂yk-∂yk- Ukl ≈ ∂xi ∂xj Vij i,j ⃗x=⃗μ
(1.24)

Error propagation for sum of uncorrelated variables:

2 2 2 σy = σ 1 + σ 2 + 2cov [x1,x2 ]
(1.25)

Error propagation for product of uncorrelated variables:

2 2 2 σ-y= σ1-+ σ2-+ 2cov[x1,x2] y2 x21 x22 x1x2
(1.26)

Characteristic function:

ikx ∫ ∞ ikx ϕx(k) = E [e ] = e f(x)dx -∞
(1.27)

Moments of Characteristic function:

dm ---m ϕz(k) = imμ′m dk
(1.28)

1.4 Important distributions

1.4.1 Binomial distribution

Characteristic function:

[( ik ) ]N ϕp (k ) = p e - 1 + 1
(1.29)

Distribution:

----N-!---- n N -n f(n; N,p ) = n!(N - n )!p (1 - p )
(1.30)

Expectation value:

E [n ] = N p
(1.31)

Variance:

V [n] = N p(1 - p)
(1.32)

1.4.2 Poisson distribution

Characteristic function:

ν(eik-1) ϕν(k) = e
(1.33)

Distribution:

νn-ν f(n;ν ) = n!e
(1.34)

Expectation value:

E [n ] = ν
(1.35)

Variance:

V [n ] = ν
(1.36)

1.4.3 Uniform distribution

Characteristic function:

eiβk - eiαk ϕα,β(k) = (β---α-)ik
(1.37)

Distribution:

1 f (x;α,β ) = ------f or α ≤ x ≤ β β - α
(1.38)

Expectation value:

α + β E [x] = --2---
(1.39)

Variance:

2 V [x] = (β --α)-- 12
(1.40)

1.4.4 Exponential distribution

Characteristic function:

1 ϕ ξ(k ) = -------- 1 - ik ξ
(1.41)

Distribution:

1 -x∕ξ f(x;ξ ) = -e for x ≥ 0 ξ
(1.42)

Expectation value:

E [x ] = ξ
(1.43)

Variance:

2 V [x] = ξ
(1.44)

1.4.5 Gaussian distribution

ú Characteristic function:

iμk- 12σ2k2 ϕμ,σ(k) = e
(1.45)

Distribution:

( 2) f(x; μ,σ) = √--1-- exp - (x---μ)-- 2π σ 2σ2
(1.46)

Expectation value:

E [x] = μ
(1.47)

Variance:

V [x] = σ2
(1.48)

1.4.6 Multivariate Gaussian distribution

Distribution:

1 ( 1 ) f(⃗x;m⃗u, V ) = ----n∕2---1∕2 exp - -(⃗x - ⃗μ)tV -1(⃗x - ⃗μ) (2π) |V| 2
(1.49)

1.4.7 Chi-square distribution

Characteristic function:

ϕ (k ) = (1 - 2ik)- n2 n
(1.50)

Distribution:

1 f (z;n) = -----------zn∕2-1e-z∕2 2n∕2Γ (n ∕2)
(1.51)

Expectation value:

E [z] = n
(1.52)

Variance:

V [z] = 2n
(1.53)

1.4.8 Cauchy (Breit-Wigner) distribution

Characteristic function:

Γ ϕx0,Γ (k ) = e- ikx0-|k|2
(1.54)

Distribution:

1-------Γ ∕2------ f (x;Γ ,x0) = πΓ 2∕4 + (x - x )2 0
(1.55)

1.4.9 Student’s t distribution

Distribution:

( ) Γ ν+1 ( x2) -(ν+21) f (x;ν) = √-----2----- 1 + --- νπ Γ (ν∕2) ν
(1.56)

Expectation value:

E [x ] = 0
(1.57)

Variance:

ν V[x] = ------ ν - 2
(1.58)