39. Probability

1

39. Probability
Revised August 2021 by G. Cowan (RHUL).

39.1 General
Further discussion of probability can be found, e.g., in Refs. [1–8].
An abstract definition of probability can be given by considering a set S, called the sample
space, and possible subsets A, B, . . . , the interpretation of which is left open. The probability P isa real-valued function defined by the following axioms due to Kolmogorov [9]:1. For every subset A in S, P (A) ≥ 0;
2. For disjoint subsets (i.e., A ∩ B = ∅), P (A ∪ B) = P (A) + P (B);
3. P (S) = 1.
In addition, one defines the conditional probability P (A|B) (read as P of A given B) as
P (A|B) =P (A ∩ B).
P (B)(39.1)

From this definition and using the fact that A ∩ B and B ∩ A are the same, one obtains Bayes’
theorem,P (B|A)P (A)P (A|B) =.
(39.2)P (B)
From the three axioms of probability and the definition of conditional probability, one obtains thelaw of total probability,X
P (B|Ai )P (Ai ) ,(39.3)
P (B) =
ifor any subset B and for disjoint Ai with ∪i Ai = S. This can be combined with Bayes’ theorem(Eq. (39.2)) to giveP (B|A)P (A)P (A|B) = P,(39.4)i P (B|Ai )P (Ai )where the subset A could, for example, be one of the Ai .
The most commonly used interpretation of the elements of the sample space are outcomes of arepeatable experiment. The probability P (A) is assigned a value equal to the limiting frequency ofoccurrence of A. This interpretation forms the basis of frequentist statistics.
The elements of the sample space might also be interpreted as hypotheses, i.e., statements thatare either true or false, such as ‘The mass of the W boson lies between 80.3 and 80.5 GeV.’ Upon
repetition of a measurement, however, such statements are either always true or always false, i.e.,the corresponding probabilities in the frequentist interpretation are either 0 or 1. Using subjective
probability, however, P (A) is interpreted as the degree of belief that the hypothesis A is true.
Subjective probability is used in Bayesian (as opposed to frequentist) statistics. Bayes’ theorem
can be written
P (theory|data) ∝ P (data|theory)P (theory) ,(39.5)
where ‘theory’ represents some hypothesis and ‘data’ is the outcome of the experiment. HereP (theory) is the prior probability for the theory, which reflects the experimenter’s degree of belief
before carrying out the measurement, and P (data|theory) is the probability to have gotten the data
actually obtained, given the theory, which is also called the likelihood.
Bayesian statistics provides no fundamental rule for obtaining the prior probability, which
may depend on previous measurements, theoretical prejudices, etc. Once this has been specified,
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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39. Probability

2

however, Eq. (39.5) tells how the probability for the theory must be modified in the light of the newdata to give the posterior probability, P (theory|data). As Eq. (39.5) is stated as a proportionality,
the probability must be normalized by summing (or integrating) over all possible hypotheses.

39.2 Random variables
A random variable is a numerical characteristic assigned to an element of the sample space. In
the frequency interpretation of probability, it corresponds to an outcome of a repeatable experiment.
Let x be a possible outcome of an observation. If x can take on any value from a continuous range,
we write f (x; θ)dx as the probability that the measurement’s outcome lies between x and x + dx.
The function f (x; θ) is called the probability density function (p.d.f.), which may depend on one ormore parameters θ. If x can take on only discrete values (e.g., the non-negative integers), then weuse f (x; θ) to denote the probability to find the value x. In the following the term p.d.f. is often
taken to cover both the continuous and discrete cases, although technically the term density should
only be used in the continuous case.
The p.d.f. is always normalized to unity. Both x and θ may have multiple components and arethen often written as vectors. If θ is unknown, we may wish to estimate its value from a given setof measurements of x; this is a central topic of statistics (see Sec. 40).
The cumulative distribution function F (a) is the probability that x ≤ a:
Z a

f (x) dx .

F (a) =

(39.6)−∞

Here and below, if x is discrete-valued, the integral is replaced by a sum. The endpoint a isexpressly included in the integral or sum. Then 0 ≤ F (x) ≤ 1, F (x) is nondecreasing, andP (a < x ≤ b) = F (b) − F (a). If x is discrete, F (x) is flat except at allowed values of x, where it
has discontinuous jumps equal to f (x).
Any function of random variables is itself a random variable, with (in general) a different p.d.f.
The expectation value of any function u(x) isZ ∞u(x) f (x) dx ,E[u(x)] =(39.7)−∞

assuming the integral is finite. The expectation value is linear, i.e., for any two functions u and vof x and constants c1 and c2 , E[c1 u + c2 v] = c1 E[u] + c2 E[v].
The nth moment of a random variable x isnαn ≡ E[x ] =Z ∞xn f (x) dx ,(39.8a)−∞and the nth central moment of x (or moment about the mean, α1 ) isnmn ≡ E[(x − α1 ) ] =Z ∞−∞(x − α1 )n f (x) dx .

(39.8b)

The most commonly used moments are the mean µ and variance σ 2 :µ ≡ α1 ,2(39.9a)2σ ≡ V [x] ≡ m2 = α2 − µ .

(39.9b)

The mean is the location of the “center of mass” of the p.d.f., and the variance is a measure of
the square of its width. Note that V [cx + k] = c2 V [x]. It is often convenient to use the standard
deviation of x, σ, defined as the square root of the variance.
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39. Probability3

Any odd moment about the mean is a measure of the skewness of the p.d.f. The simplest ofthese is the dimensionless coefficient of skewness γ1 = m3 /σ 3 .
The fourth central moment m4 provides a convenient measure of the tails of a distribution.
For the Gaussian distribution (see Sec. 39.4), one has m4 = 3σ 4 . The kurtosis is defined as
γ2 = m4 /σ 4 − 3, i.e., it is zero for a Gaussian, positive for a leptokurtic distribution with longer
tails, and negative for a platykurtic distribution with tails that die off more quickly than those ofa Gaussian.
The quantile xα is the value of the random variable x at which the cumulative distribution
is equal to α. That is, the quantile is the inverse of the cumulative distribution function, i.e.,xα = F −1 (α). An important special case is the median, xmed , defined by F (xmed ) = 1/2, i.e.,
half the probability lies above and half lies below xmed . (More rigorously, xmed is a median if
P (x ≥ xmed ) ≥ 1/2 and P (x ≤ xmed ) ≥ 1/2. If only one value exists, it is called ‘the median.’)Under a monotonic change of variable x → y(x), the quantiles of a distribution (and hence also
the median) obey yα = y(xα ). In general the expectation value and mode (most probable value) of
a distribution do not, however, transform in this way.
Let x and y be two random variables with a joint p.d.f. f (x, y). The marginal p.d.f. of x (thedistribution of x with y unobserved) isZ ∞f1 (x) =f (x, y) dy ,(39.10)−∞and similarly for the marginal p.d.f. f2 (y). The conditional p.d.f. of y given fixed x (with f1 (x) 6= 0)
is defined by f3 (y|x) = f (x, y)/f1 (x), and similarly f4 (x|y) = f (x, y)/f2 (y). From these, we
immediately obtain Bayes’ theorem (see Eqs. (39.2) and (39.4)),f4 (x|y) =The mean of x is

f3 (y|x)f1 (x)f3 (y|x)f1 (x)=R.
f2 (y)
f3 (y|x0 )f1 (x0 ) dx0

Z ∞ Z ∞

µx =(39.11)Z ∞x f (x, y) dx dy =x f1 (x) dx ,(39.12)cov[x, y] = E[(x − µx )(y − µy )] = E[xy] − µx µy .

(39.13)−∞ −∞−∞and similarly for y. The covariance of x and y is

A dimensionless measure of the covariance of x and y is given by the correlation coefficient,ρxy = cov[x, y]/σx σy ,(39.14)

where σx and σy are the standard deviations of x and y. It can be shown that −1 ≤ ρxy ≤ 1.
Two random variables x and y are independent if and only if
f (x, y) = f1 (x)f2 (y) .

(39.15)

If x and y are independent, then ρxy = 0; the converse is not necessarily true. If x and y are
independent, E[u(x)v(y)] = E[u(x)]E[v(y)], and V [x + y] = V [x] + V [y]; otherwise, V [x + y] =V [x] + V [y] + 2cov[x, y], and E[uv] does not necessarily factorize.
Consider a set of n continuous random variables x = (x1 , . . . , xn ) with joint p.d.f. f (x), and aset of n new variables y = (y1 , . . . , yn ), related to x by means of a function y(x) that is one-to-one,i.e., the inverse x(y) exists. The joint p.d.f. for y is given byg(y) = f (x(y))|J| ,31st May, 2024

(39.16)

39. Probability

4where |J| is the absolute value of the determinant of the square matrix Jij = ∂xi /∂yj (the Jacobian
determinant). If the transformation from x to y is not one-to-one, the x-space must be broken into
regions where the function y(x) can be inverted, and the contributions to g(y) from each region
summed.
Given a set of functions y = (y1 , . . . , ym ) with m < n, one can construct n − m additional independent functions, apply the procedure above, then integrate the resulting g(y) over the unwanted
yi to find the marginal distribution of those of interest.
For a one-to-one transformation of discrete random variables, the probability is obtained bysimple substitution; no Jacobian is necessary because in this case f is a probability rather than aprobability density. If the transformation is not one-to-one, then one must sum the probabilities
for all values of the original variable that contribute to a given value of the transformed variable. If
f depends on a set of parameters θ, a change to a different parameter set η(θ) is made by simple
substitution; no Jacobian is used.
39.2.1 Propagation of errors
Consider n random variables x = (x1 , . . . , xn ) and m functions y(x) = (y1 (x), . . . , ym (x)).
Suppose here that the mean values µ = (µ1 , . . . , µn ) = E[x] are known, although in practicethey will only be estimated, and suppose we also know or have estimated the covariance matrix
Vij = cov[xi , xj ]. The goal of error propagation is to determine the covariance matrix for thefunctions, Uij = cov[yi , yj ]. In particular, the diagonal elements Uii = V [yi ] give the variances. The
new covariance matrix can be found by expanding the functions y(x) about the means µ to firstorder in a Taylor series. Using this one finds
Uij ≈X ∂yi ∂yjk,l∂xk ∂xl µVkl .

(39.17)This can be written in matrix notation as U ≈ AV AT where the matrix of derivatives A is
Aij =

∂yi
,
∂xj µ

(39.18)and AT is its transpose. The approximation is exact if y(x) is linear. If this is not the case, theapproximation can break down if, for example, y(x) is significantly nonlinear close to µ in a regionof a size comparable to the standard deviations of x.

39.3 Characteristic functions
The characteristic function φ(u) associated with the p.d.f. f (x) is essentially its Fourier transform, or the expectation value of eiux :hiφ(u) = E eiux =Z ∞eiux f (x) dx .

(39.19)−∞Once φ(u) is specified, the p.d.f. f (x) is uniquely determined and vice versa; knowing one isequivalent to the other. Characteristic functions are useful in deriving a number of importantresults about moments and sums of random variables.
It follows from Eqs. (39.8) and (39.19) that the nth moment of a random variable x that follows
f (x) is given byZ ∞n−n d φi=xn f (x) dx = αn .
(39.20)dun u=0−∞Thus it is often easy to calculate all the moments of a distribution defined by φ(u), even when f (x)cannot be written down explicitly.
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If the p.d.f.s f1 (x) and f2 (y) for independent random variables x and y have characteristic functions φ1 (u) and φ2 (u), then the characteristic function of the weighted sum ax + by is φ1 (au)φ2 (bu).
The rules of addition for several important distributions (e.g., that the sum of two Gaussian distributed variables also follows a Gaussian distribution) easily follow from this observation.
Let the (partial) characteristic function corresponding to the conditional p.d.f. f2 (x|z) be
φ2 (u|z), and the p.d.f. of z be f1 (z). The characteristic function after integration over the conditional value isZ
φ(u) =

φ2 (u|z)f1 (z) dz .

(39.21)

Suppose we can write φ2 in the form
φ2 (u|z) = A(u)eig(u)z .

(39.22)φ(u) = A(u)φ1 (g(u)) .

(39.23)Then

The cumulants (semi-invariants) κn of a distribution with characteristic function φ(u) are defined by the relation
φ(u) = exp

"∞
X κn
n=1

n!

#
n

(iu)

1
= exp iκ1 u − κ2 u2 + . . .
2




.

(39.24)The values κn are related to the moments αn and mn . The first few relations areκ1 = α1 (= µ, the mean)
κ2 = m2 = α2 − α12 (= σ 2 , the variance)(39.25)

κ3 = m3 = α3 − 3α1 α2 + 2α13 .

39.4 Commonly used probability distributions
Table 39.1 gives a number of common probability density functions and corresponding characteristic functions, means, and variances. Further information may be found in Refs. [1–8] [10],and [11], which has particularly detailed tables. Monte Carlo techniques for generating each ofthem may be found in our Sec. 42.4 and in Ref. [10]. We comment below on all except the trivial
uniform distribution.
39.4.1 Binomial and multinomial distributions
A random process with exactly two possible outcomes which occur with fixed probabilities
is called a Bernoulli process. If the probability of obtaining a certain outcome (a “success”) in
an individual trial is p, then the probability of obtaining exactly r successes (r = 0, 1, 2, . . . , N )in N independent trials, without regard to the order of the successes and failures, is given by thebinomial distribution f (r; N, p) in Table 39.1. If r and s are binomially distributed with parameters(Nr , p) and (Ns , p), then t = r + s follows a binomial distribution with parameters (Nr + Ns , p). If
there are are m possible outcomes for each trial having probabilities p1 , p2 , . . . , pm , then the joint
probability to find r1 , r2 , . . . , rm of each outcome after a total of N independent trials is given by
the multinomial distribution as shown in Table 39.1. We can regard outcome i as “success” and all
the rest as “failure”, so individually, any of the ri follow a binomial distribution for N trials and a
success probability pi .
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39.4.2 Poisson distribution
The Poisson distribution f (n; ν) gives the probability of finding exactly n events in a given
interval of x (e.g., space or time) when the events occur independently of one another and of x at
an average rate of ν per the given interval. The variance σ 2 equals ν. It is the limiting case p → 0,
N → ∞, N p = ν of the binomial distribution. The Poisson distribution approaches the Gaussian
distribution for large ν.
For example, a large number of radioactive nuclei of a given type will result in a certain numberof decays in a fixed time interval. If this interval is small compared to the mean lifetime, then theprobability for a given nucleus to decay is small, and thus the number of decays in the time interval
is well modeled as a Poisson variable.
39.4.3 Normal or Gaussian distribution
The normal (or Gaussian) probability density function f (x; µ, σ 2 ) given in Table 39.1 has mean
E[x] = µ and variance V [x] = σ 2 . Comparison of the characteristic function φ(u) given in Table 39.1
with Eq. (39.24) shows that all cumulants κn beyond κ2 vanish; this is a unique property of theGaussian distribution. Some other properties are:P (x in range µ ± σ) = 0.6827,
P (x in range µ ± 0.6745σ) = 0.5,pE[|x − µ|] = 2/πσ = 0.7979σ,√
half-width at half maximum = 2 ln 2 σ = 1.177σ.
For a Gaussian with µ = 0 and σ 2 = 1 (the standard normal) the cumulative distribution, often
written Φ(x), is related to the error function erf by
F (x; 0, 1) ≡ Φ(x) =√ i
1h
1 + erf(x/ 2) .
2

(39.26)The error function and standard Gaussian are tabulated in many references (e.g., Ref. [11,12]) andare available in software packages such as ROOT [13]. For a mean µ and variance σ 2 , replace x by
(x − µ)/σ. The probability of x in a given range can be calculated with Eq. (40.76).
For x and y independent and normally distributed, z = ax+by follows a normal p.d.f. f (z; aµx +bµy , a2 σx2 + b2 σy2 ); that is, the weighted means and variances add.
The Gaussian derives its importance in large part from the central limit theorem:
If independent random variables x1 , . . . , xn are distributed according to any p.d.f. with finitePmean and variance, then the sum y = ni=1 xi will have a p.d.f. that approaches a Gaussian forlarge n. If the p.d.f.s of the xi are not identical, the theorem still holds under somewhat more
restrictive conditions. The mean and variance are given by the sums of corresponding terms fromthe individual xi . Therefore, the sum of a large number of fluctuations xi will be distributed as a
Gaussian, even if the xi themselves are not.
For a set of n Gaussian random variables x with means µ and covariances Vij = cov[xi , xj ], thep.d.f. for the one-dimensional Gaussian is generalized to11pf (x; µ, V ) =exp − (x − µ)T V −1 (x − µ) ,
n/2
2
(2π)
|V |




(39.27)

where the determinant |V | must be greater than 0. For diagonal V (independent variables),f (x; µ, V ) is the product of the p.d.f.s of n Gaussian distributions.
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39. Probability

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For n = 2, f (x; µ, V ) isf (x1 , x2 ; µ1 , µ2 , σ1 , σ2 , ρ) =
(

× exp

1
p

2πσ1 σ2 1 − ρ2"−1(x2 − µ2 )2(x1 − µ1 )22ρ(x1 − µ1 )(x2 − µ2 )+−2(1 − ρ2 )
σ1 σ2
σ12
σ22

(39.28)#).

The characteristic function for the multivariate Gaussian is1
φ(u; µ, V ) = exp iµ · u − uT V u .
2(39.29)If the components of x are independent, then Eq. (39.29) is the product of the characteristic
functions of n Gaussians.
For an n-dimensional Gaussian distribution for x with mean µ and covariance matrix V , themarginal distribution for any single xi is is a one-dimensional Gaussian with mean µi and variance
Vii . The equation (x − a)T V −1 (x − a) = C, where C is any positive number, defines an ndimensional ellipse centered about a. If a is equal to the mean µ, then C is a random variable
obeying the χ2 distribution for n degrees of freedom, which is discussed in the following section.
The probability that x lies outside the ellipsoid for a given value of C is given by 1 − Fχ2 (C; n),where Fχ2 is the cumulative χ2 distribution. This may be read from Fig. 40.1. For example, the“s-standard-deviation ellipsoid” occurs at C = s2 . For the two-variable case (n = 2), the point x
lies outside the one-standard-deviation ellipsoid with 61% probability. The use of these ellipsoids
as indicators of probable error is described in Sec. 40.4.2.2; the validity of those indicators assumes
that µ and V are correct.
39.4.4 Log-normal distribution
If a random variable y follows a Gaussian distribution with mean µ and variance σ 2 , thenx = ey follows a log-normal distribution, as given in Table 39.1. As a consequence of the central
limit theorem described in Sec. 39.4.3, the distribution of the product of a large number of positive
random variables approaches a log-normal. It is bounded below by zero and is thus well suited formodeling quantities that are intrinsically non-negative such as an efficiency. One can implement
a log-normal model for a random variable x by defining y = ln x so that y follows a Gaussian
distribution.
39.4.5 χ2 distributionP
If x1 , . . . , xn are independent Gaussian random variables, the sum z = ni=1 (xi −µi )2 /σi2 follows
the χ2 p.d.f. with n degrees of freedom, which we denote by χ2 (n). More generally, for n correlated
Gaussian variables as components of a vector X with covariance matrix V , z = X T V −1 X follows
PP
χ2 (n) as in the previous section. For a set of zi , each of which follows χ2 (ni ), zi follows χ2 ( ni ).
For large n, the χ2 p.d.f. approaches a Gaussian with a mean and variance given by µ = n andσ 2 = 2n, respectively (here the formulae for µ and σ 2 are valid for all n).
The χ2 p.d.f. is often used in evaluating the level of compatibility between observed data anda hypothesis for the p.d.f. that the data might follow. This is discussed further in Sec. 40.3.2 on
significance tests.
39.4.6 Student’s t distribution
Suppose that y and x1 , . . . , xn are independent and Gaussian distributed with mean 0 andvariance 1. We then definenXyz=x2i and t = p.
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39. Probability

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Table 39.1: Some common probability density functions, with corresponding characteristic functions and means and variances. In the Table,Γ (k) is the gamma function, equal to (k − 1)! when k is an integer; 1 F1 isthe confluent hypergeometric function of the 1st kind [11].

Distribution
Uniform
Binomial

Probability density function
f (variable; parameters)(
1/(b − a)a≤x≤bf (x; a, b) =0otherwise!
f (r; N, p) = r!(NN−r)!
pr q N −rr = 0, 1, 2, . . . , N ; 0 ≤ p ≤ 1; q = 1 − p

Mean

Variance(b−a)iua+b2(b−a)212(q + peiu )N

Np

N pq
cov[ri , rj ] =N pi (δij − pj )νeibu −eiauef (n; ν) = ν n!
; n = 0, 1, 2, . . . ; ν > 0

exp[ν(eiu − 1)]E[ri ] =N piν

f (x; µ, σ 2 ) = σ √12π exp(−(x − µ)2 /2σ 2 )exp(iµu − 21 σ 2 u2 )

µ

σ2

u

Vjk

N!
Multinomial f (r1 , . . . , rm ; N, p1 , . . . , pm ) = r1 !···r
pr1 · · · prmm
m! 1

PoissonNormal(Gaussian)Multivariate
Gaussian

Characteristic
function φ(u)n −νf (x; µ, V ) =1√(2π)n/2Pm

k=1 pk e


iuk N

h

i

exp iµ · u − 21 uT V u

|V |

i

h

× exp − 12 (x − µ)T V −1 (x − µ)−∞ < xj < ∞; − ∞ < µj < ∞; |V | > 0
Log-normal
χ2Student’s t
Gamma
Beta

exp(µ + σ 2 /2)

exp(2µ + σ 2 )×[exp(σ 2 ) − 1]f (x; µ, σ 2 ) = σ √12π x1 exp(−(ln x − µ)2 /2σ 2 )—0 < x < ∞; −∞ < µ < ∞; σ > 0
n/2−1 e−z/2; z≥0f (z; n) = z2n/2 Γ (n/2)(1 − 2iu)−n/2

n

2n

—

0
for n > 1n/(n − 2)for n > 2

k/λ

k/λ2

f (t; n) = √1nπ Γ [(n+1)/2]
Γ (n/2)
2 −(n+1)/2

1 + tn−∞ < t < ∞;n not required to be integer
k−1 k −λxf (x; λ, k) = x Γλ(k)e ; 0 ≤ x < ∞ ;(1 − iu/λ)−kk not required to be integer(α+β) α−1f (x; α, β) = ΓΓ(α)Γ
(1 − x)β−1
1 F1 (α; α + β; iu)
(β) x
0≤x≤1

α
α+β

αβ
(α+β)2 (α+β+1)

The variable z thus follows a χ2 (n) distribution. Then t is distributed according to Student’s t
distribution with n degrees of freedom, f (t; n), given in Table 39.1.
If defined through gamma functions as in Table 39.1, the parameter n is not required to be
an integer. As n → ∞, the distribution approaches a Gaussian, and for n = 1 it is a Cauchy or
Breit–Wigner distribution.
PP
As an example, consider the sample mean x = xi /n and the sample variance s2 = (xi −x)2 /(n − 1) for normally distributed xi with unknown mean µ and variancep
σ 2 . The sample mean
2
has a Gaussian distribution with a variance σ /n, so the variable (x − µ)/ σ 2 /n is normal withmean 0 and variance 1. The quantity (n − 1)s2 /σ 2 is independent of this and follows χ2 (n − 1).
The ratiopx−µ(x − µ)/ σ 2 /npt= p=(n − 1)s2 /σ 2 (n − 1)s2 /n

(39.31)is distributed as f (t; n − 1). The unknown variance σ 2 cancels, and t can be used to test the
hypothesis that the true mean is some particular value µ.
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939. Probability

39.4.7 Gamma distribution
For a process that generates events as a function of x (e.g., space or time) according to a Poisson
distribution, the distance in x from an arbitrary starting point (which may be some particular event)to the k th event follows a gamma distribution, f (x; λ, k). The Poisson parameter µ is λ per unit x.
The special case k = 1 (i.e., f (x; λ, 1) = λe−λx ) is called the exponential distribution. A sum of k 0
Pexponential random variables xi is distributed as f ( xi ; λ, k 0 ).
The parameter k is not required to be an integer. For λ = 1/2 and k = n/2, the gamma
distribution reduces to the χ2 (n) distribution.
39.4.8 Beta distribution
The beta distribution describes a continuous random variable x in the interval [0, 1]. By scaling
and translation one can easily generalize it to have arbitrary endpoints. In Bayesian inference about
the parameter p of a binomial process, if the prior p.d.f. is a beta distribution f (p; α, β) then theobservation of r successes out of N trials gives a posterior beta distribution f (p; r + α, N − r + β)(Bayesian methods are discussed further in Sec. 40). The uniform distribution is a beta distribution
with α = β = 1.
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[2] A. Stuart and J.K. Ord, Kendall’s Advanced Theory of Statistics, Vol. 1 Distribution Theory
6th Ed., (Halsted Press, New York, 1994), and earlier editions by Kendall and Stuart.
[3] F.E. James, Statistical Methods in Experimental Physics, 2nd Ed., (World Scientific, Singapore,2006).
[4] L. Lyons, Statistics for Nuclear and Particle Physicists, (Cambridge University Press, New
York, 1986).
[5] B.R. Roe, Probability and Statistics in Experimental Physics, 2nd Ed., (Springer, New York,
2001).
[6] R.J. Barlow, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences,(John Wiley, New York, 1989).
[7] S. Brandt, Data Analysis, 3rd Ed., (Springer, New York, 1999).
[8] G. Cowan, Statistical Data Analysis, (Oxford University Press, Oxford, 1998).
[9] A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, (Springer, Berlin, 1933);
Foundations of the Theory of Probability, 2nd Ed., (Chelsea, New York 1956).
[10] Ch. Walck, Hand-book on Statistical Distributions for Experimentalists, University of Stockholm Internal Report SUF-PFY/96-01, available from www.physto.se~walck.
[11] M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, (Dover, New York,1972).
[12] F.W.J. Olver et al., eds., NIST Handbook of Mathematical Functions, (Cambridge University
Press, 2010); A companion Digital Library of Mathematical Functions is available at dlmf.
nist.gov.
[13] R. Brun and F. Rademakers, Nucl. Instrum. Meth. A389, 81 (1997); See also root.cern.ch.

31st May, 2024