The birth process is a simple continuous-time Markov process. It models the conti-nuous production of a species S at a birth rate c. The model is made up of a synthesisreaction:
R1 : /0c→ S.
The state of the system is the current size of population of species S. When a birthoccurs, the population of species S is increased by 1.
A.2 Fast Isomerization Model
The fast isomerization model describes a simple reaction model with stiffness. Thecore of the model is a pair of reversible isomerization reactions that transform amolecule into another molecule. The isomerization process is assumed to occur at arate much faster than that of other reactions in the model. The isomerization processis thus quickly reaching a partial equilibrium and species involved in these reactionsapproach quasi-steady state.
Table A.1 lists the reactions in the fast isomerization model. It contains threespecies involved in three reactions. The first two reactions represent the conversionback and forth of the species S1 and S2 at rates k1 and k2, respectively. The reactionR3 denotes a slow reaction that transforms species S2 to S3 at rate k3 such thatk1,k2 � k3. In this model, the mass-action kinetics is applied for each reaction.
The Oscillator [58] is a noise-induced system. The basic component of the modelis a positive feedback loop that enhances the production rates of reactions. The re-action products are therefore sharply increasing. The result of the Oscillator is asymmetrical bell shape behavior in the dynamics of each molecular species.
The model consists of three species involved in three reactions listed in Table A.2.Each reaction transforms a two reactant species into two copies of one of the tworeactants implementing the positive feedback process. For instance, reaction R1 willtransform a species A into species B that in turn catalyzes and positively amplifiesthe reaction rate. The emergence of the positive feedback loop has two mutual regu-latory effects on the behavior of reaction R1: it increases the transformation rate of Ainto B and the increasing amount of species B in the system boosts up the reaction.The species B is thus produced more and more approaching the saturation.
Table A.2 Oscillator model
Reactions
R1: A + B → 2B R2: B + C → 2C R3: C + A → 2A
The initial populations of species in the model are: #A= 900, #B= 500 and #C =200. The mass-action kinetics is applied for each reaction with their rate constants:c1 = c2 = c3 = 1.
A.4 Schlogl Model
The Schlogl model [229] is a reaction network which can exhibit bistability andswitching behaviour [108, 176, 270]. The system has two stable steady states sepa-rated by an unstable state. In the deterministic framework, the system after a shorttime will converge to one of the steady states and reside in this steady state de-pending on where its basin of attraction closing to the initial condition. The systemdynamics in the stochastic framework may jump between the two stable states spon-taneously, due to its inherent randomness. Such a behavior is referred to as stochas-tic switching [267] because it cannot be observed in the deterministic framework.
A.5 Oregonator Model 209
The reactions of the Schlogl model are listed in Table A.3. The model consists ofthree species and four reactions implementing a trimolecular, autocatalytic reactionscheme. The population of species A and B are highly abundant and assumed to beconstant values (buffer molecules). The state of the system is thus represented bythe number of species X.
Table A.3 Schlogl model
Reaction
R1: A + 2X → 3X R2: 3X → A + 2X
R3: B → X R4: X → B
The initial condition of species is #X = 250, #A = 1.0e5 and #B = 2.0e5. Therate constants of reactions are c1 = 3.0e–7, c2 = 1.0e–4, c3 = 1.0e–3 and c4 = 3.5.
A.5 Oregonator Model
The Oregonator model [86] is the simplest realistic model of the nonlinear oscil-latory Belousov-Zhabotinsky (BZ) reaction [285, 31, 79]. The model is importantbecause it exhibits oscillating phenomena even if the system is far from the ther-modynamic equilibrium [87]. Extensions of the basic Oregonator model have beenproposed to investigate complex patterns such as traveling waves in the reaction-diffusion [22, 193, 194]. The Oregonator is of interest for both theoretical researchand practical simulation.
Here we will consider a simplified version of the oscillator involving three spe-cies and five reactions listed in Table A.4. The underlying mechanism of the Orego-nator is regulated by an autocatalytic reaction (reaction R3) and a negative feedbackloop (via reactions R3, R5 and R1). The products of reaction R3 are the activator Xand the inhibitor Z. The activator X catalyzes back the production of its reaction(hence, the name autocalytic reaction). The inhibitor Z inhibits the autocatalyticproduction of X through the reaction sequence R3 → R5 → R1. The presence ofboth the activator and inhibitor processes causes a nonlinear behavior leading to thespontaneous generation of oscillations.
The initial condition of species is assigned to #X = 500, #Y = 1,000, #Z = 2,100.The mass-action stochastic kinetics rates are: c1 = 0.1, c2 = 2, c3 = 104, c4 = 0.016,c5 = 26.
210 A Benchmark Models
Table A.4 Oregonator model
Reaction
R1: X + Y → /0 R2: Y → X
R3: X → 2X + Z R4: 2X → /0
R5: Z → Y
A.6 Gene Expression Model
Gene expression is an important regulatory process which transcribes and transla-tes the genetic information encoded in DNA, referred to as gene, into functionalgene products, often proteins. Proteins play a vital role in many cellular functions.Therefore, gene expression is the fundamental process that translates the genotypeinto the phenotype [265, 215, 134]. The transcription occurs when RNA-polymerase(RNAP) binds to the promoter region of the gene. During the transcription process,the gene is copied to intermediate form called messenger RNA (mRNA). mRNAthen binds to ribosomes to translate into the corresponding protein. The role of geneexpression in biological systems have been studied intensively not only becauseit affects the behavior of cellular processes, but also for its inherent stochasticity,which is known as biological noise [179, 180, 200, 272, 21, 77, 78, 214, 140]. Thereare two sources of noise. The first source of noise comes from the low copy num-bers of regulatory molecules involved in this process which results in the proteinsbeing produced as a bursting wave rather than a continuous wave [85]. This type ofnoise is called intrinsic noise. The second type of noise is called extrinsic noise. Theextrinsic noise is caused by environmental factors. For example, Bratsun et al. [38]showed that delay in the degradation of proteins in the gene expression can leadto oscillation. Another example is the exposing of switch-like behavior in the geneexpression if the protein activation process is modelled by a Hill kinetics instead ofmass-action kinetics [142].
A prototypical gene expression model with five species and eight reactions isdepicted in Table A.5. Protein P is encoded by its gene G. The intermediate productof transcription is denoted by M. The transcription was modeled by reaction R1
where gene G transcribes to M. The M translates to protein P in reaction R2 ordegrades by reaction R3. Two proteins P interact to form a reversible dimer P2 (R5,R6) or degrade (R4). The dimer could bind to gene G to enhance the activation ofthe gene (R7,R8).
The initial condition of the gene expression model is: #G = 1,000, and #M =#P = #P2 = #P2G = 0. The stochastic rates for reactions are: c1 = 0.09, c2 = 0.05,c3 = 0.001, c4 = 0.0009, c5 = 0.00001, c6 = 0.0005, c7 = 0.005 and c8 = 0.9.
A.7 Folate Cycle Model 211
Table A.5 Gene expression model
Reaction
R1: G → G+M R2: M → M+P
R3: M → /0 R4: P → /0
R5: 2P → P2 R6: P2 → 2P
R7: P2 +G → P2G R8: P2G → P2 +G
A.7 Folate Cycle Model
The folate cycle is a metabolic pathway which has a vital role in cell metabo-lism [25, 195]. It transfers one-carbon units for methylation to produce methionineand synthesis of pyrimidines and purines.
The model consists of seven species and 13 reactions listed in Table A.6. Thefolate cycle begins when folic acid is reduced to tetrahydrofolate (THF). THF iscatalysed by serine hydroxymethyl transferase (SHMT) to produce 5,10-methylene-THF. Then, 5,10-methylene-THF is either converted to 5-methyl-THF catalysed byenzyme methylenetetrahydrofolate reductase (MTHFR) or converted to 10-formyl-THF. The folate cycle completes when 5-methyl-THF is demethylated to producemethionine and THF.
Table A.6 Folate cycle model
Reaction
R1: THF → 5,10-methylene-THF
R2: 5,10-methylene-THF → THF
R3: 5,10-methylene-THF → DHF
R4: DHF → THF
R5: THF → 10-formyl-THF
R6: 10-formyl-THF → THF
R7: 10-formyl-THF → THF
R8: 10-formyl-THF → 5,10-methenyl-THF
R9: 5,10-methenyl-THF → 10-formyl-THF
R10: 5,10-methenyl-THF → 5,10-methylene-THF
R11: 5,10-methylene-THF → 5,10-methenyl-THF
R12: 5,10-methylene-THF → 5-methyl-THF
R13: 5-methyl-THF → THF + Methionine
The initial abundance of species is: #THF = 8,157, #10-formyl-THF = 31,338,#5,10-methylene-THF = 3,688, #5,10-methenyl-THF = 10,244, #DHF = 87, #5-methyl-THF = 19,842 and #Methionine = 0. All reactions in the folate cycle are
212 A Benchmark Models
enzymatic reactions with Michaelis-Menten kinetics [218]. The propensity of a re-action R j in the folate cycle has form:
a j =Vm
Km +XX
where X is the population of reactant. The values of the maximal rate Vm and theMichaelis constant Km are dependent on the specific reaction [218, 232] and arelisted in Table A.7.
Table A.7 Kinetics parameters for the folate cycle model
Reaction Km Vm
R1 379.39490 1.70727e8
R2 98,642.67437 5.32670e9
R3 94,848.72536 5.32670e9
R4 23,901.87879 1.90816e9
R5 7,587.89802 2.02824e9
R6 3,414.55411 2.25586e7
R7 1,138.18470 2.31852e7
R8 758.78980 2.69327e8
R9 758.78980 2.34442e8
R10 16,313.98076 1.35038e7
R11 76,258.37518 2.82920e8
R12 379.39490 8.14996e4
R13 1,896.97450 2.64940e4
A.8 MAPK Cascade Model
The mitogen-activated protein (MAP) kinase (MAPK) cascade pathway describes achain of proteins that cascade a signal from the cell receptor to its nucleus and resultin a cellular response, i.e., cell proliferation, division and apoptosis [234, 199, 64].MAPK cascade is a ubiquitously and highly conserved regulatory module pro-cessing information during the signal transduction in cells. The MAPK pathwayis stimulated when ligands, e.g., growth factors, bind to the receptor on the cellsurface. The process is controlled through three main protein kinases: MAPKKK,MAPKK and MAPK. The propagation of the stimulated signal is cascaded by se-quential phosphorylation and activations of these kinases. First, the ligand activatesMAPKKK. The activated MAPKKK in turn phosphorylates MAPKK and subse-quently activates MAPK through further phosphorylation.
A.9 FcεRI Pathway Model 213
We consider two MAPK models, a simplified MAPK model with 12 speciesand 10 reactions [197] listed in Table A.8 (with the proteins kinases represented byKKK, KK and K) and a complex one with 106 species and 296 reactions [149] (seethe reference for the reactions and kinetics parameters). A signal E1 triggers thecascade of phosphorylations that leads to the activation of a protein. An end signalE2 triggers a cascade of phosphatases that reverts back the activation of the protein.
Table A.8 MAPK model
Reaction
R1: KKK + E1 → KKKp + E1 R2: KKKp + E2 → KKK + E2
R3: KK + KKKp → KKp + KKKp R4: KKp + KKpase → KK + KKpase
The initial population of species is: #E1 = #E2 = 20,000, #Kpase = #KKpase =20,000, #K = #KK = 2,000,000, #KKK = 200,000 and all other species are zero.The kinetics rates of reactions are c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 = c9 =c10 = 1.0e−4.
A.9 FcεRI Pathway Model
The high-affinity IgE receptor, which is referred to as FcεRI, forms a high-affinitycell surface receptor for the antigen-specific immunoglobulin E (IgE). The FcεRIreceptor is a tetramer consisting of three subunits that are: an α chain (FcεRIα),a β chain (FcεIβ ), and two disulfide bridge connected γ chains (FcεRIγ). The αchain is the antibody binding site for IgE, while the others have the role of initia-ting and amplifying the downstream signaling. The crosslinking of the IgE-antigencomplex and the aggregation of the FcεRI lead to degranulation and release of al-lergic mediators from the immune system [84]. The FcεRI signaling pathway hasbeen studied extensively in the literature due to its major role in controlling allergicresponses [61].
The model [165] is created to analyze the mechanisms of Syk phosphorylation.After FcεRI aggregation, Lyn, a membrane-associated Src family protein tyrosinekinase (SFK), is activated and phosphorylates the immunoreceptor tyrosine-basedactivation motifs (ITAMs) in the β and γ subunits of FcεRI. Phosphorylated ITAMsof the β and γ subunits provide sites for the binding and activation of Syk, a cy-tosolic protein tyrosine kinase. Activated Syk then phosphorylates many substratesleading to the activation of several signaling pathways. The crafted model contains
214 A Benchmark Models
380 species and 3,862 reactions (see [165] for the list of reactions and kinetics pa-rameters).
A.10 B Cell Antigen Receptor Signaling Model
The B cell receptor (BCR) is an antigen (Ag) receptor presented on the B cell’souter surface. It has a membrane-bound immunoglobulin (Ig) and a transmembraneprotein CD79, which is composed of two disulfide-linked chains called CD79A (Ig-α) and CD79B (Ig-β ). The binding of Ags to the membrane Ig subunit stimulatesthe receptor aggregation and transmits the signals to the cell interior through theIg-α/β subunits. BCR aggregation activates Lyn and Fyn, which are the Src familyprotein tyrosine kinases (SFKs), as well as other tyrosine kinases and initiates theBCR signaling pathway. The BCR signaling in turn activates multiple signalingcascades which results in many possible effects on the fates of B cells includingproliferation, differentiation and apoptosis [203, 105, 111, 153].
The model [29] studies the effects of Lyn and Fyn redundancy on the pathway.The basis of the model includes two feedback loops. The first loop is a positivefeedback loop that emanates upon the SFK-mediated phosphorylation of BCR andreceptor-bound Lyn and Fyn. The positive feedback loop increases the kinase acti-vities of Lyn and Fyn. The second one is a negative feedback loop arising fromSFK-mediated phosphorylation of the transmembrane adapter protein PAG1 (phos-phoprotein associated with glycosphingolipid-enriched microdomains) which de-creases the kinase activities of Lyn and Fyn. This model was implemented witha rule-based modeling approach by including the site-specific details of protein-protein interactions. The reaction network generated from the model contains 1,122species and 24,388 reactions (see [29] for reactions and parameters).
A.11 Linear Chain Model
The linear chain model is an artificial model that is used to measure the scalability ofsimulation algorithms. It models the transformation of a species to another species.The number of affected reactions which need to update their propensities in theLinear chain model is fixed by 2.
The linear chain model consists of N species Si for i = 1, . . .N. The number ofreactions M in this model is equal to the number of species N, i.e., M = N, in whichthe Ri reaction transforms species Si into the species Si+1 as{
Sici→ Si+1 , for i = 1, . . .N −1
SNcN→ S1 , for i = N
where ci is the rate constant of the transformation.
A.11 Linear Chain Model 215
The kinetics rates of all reactions are set to ci = 1 for i = 1 . . .N. The initialpopulation of each species Si for i = 1 . . .N is randomly taken from 0 and 10,000.
Appendix BRandom Number Generation
The appendix recalls methods for implementing random number generators (RNG)to generate random numbers used for stochastic simulation throughout the book. Acomprehensive review of methods can be found in [148, 204, 65, 119]. An RNGis a physical or computational method to generate a sequence of (pseudo-)randomnumbers. The random numbers generated by an RNG must appear to be independentand identically distributed (i.i.d.), i.e., each random number has the same specifiedprobability distribution as the others and all are mutually independent.
B.1 Uniform Random Number Generator
The goal of a uniform random number generator is to generate sequences of uni-formly distributed random numbers in (0,1). Almost all the mainstream program-ming languages and scientific libraries provide routines for generating such sequen-ces of random numbers. Here we present only linear congruential generators (LCG)because they constitutes a milestone in the field. A LCG, however, should be usedwith care in applications where high-quality randomness is required because of theserial correlation in the output. In such cases, improved random number generatorssuch as Xorshift [174] or Mersenne Twister [177] should be preferred. In fact, Xors-hift and Mersenne Twister strategies actually rely on a LCG to generate the randomseed.
The basis of a LCG is a recurrence
Xi = (aXi−1 + c) mod m
to compute a sequence of integer values Xi and requires four integer parameters:an initial seed X0, a modulus m, a multiplier a and an increment c. In special casec = 0, the LCG is called multiplicative. The choice of parameters is important toobtain a long period (the number of outcomes before the generator start repeating thesequence) and a fast generation time. For multiplicative generators, good parameters
are m = 231 −1, a = 75 and X0 ∈ [1,m−1]. The period of the generator with theseparameters is m−1.
The sequence of random numbers ri ∼ U(0,1) is obtained by dividing the valuesXi by m,
ri = Xi/m.
Algorithm 50 implements the LCG for generating uniformly distributed randomnumbers.
Algorithm 50 Linear congruential generator (LCG)Input: integers X0, a, c, m.Output: a sequence of pseudo-random numbers independent and identically distributed in U(0,1).1: set i = 12: while (true) do3: compute Xi = (aXi−1 + c) mod m4: set random number ri = Xi/M5: set i = i+16: end while7: return random numbers ris
B.2 Non-uniform Random Number Generator
The purpose of a non-uniform random number generator is to transform a sequenceof i.i.d. random numbers of U(0,1) into a sequence of the desired distribution. Thetransformation does not need to be one-to-one. The techniques for generating non-uniform random numbers include the inversion method, the (acceptance-)rejectionmethod and the composition method.
B.2.1 General Techniques
B.2.1.1 Inversion Method
Let X be a random variable with a probability density function (pdf) f (x). Let F(x)be the cumulative distribution function (cdf) of X . The principle of the inversionmethod is based on the fact that if r is a uniform random number U(0,1), thenX = F−1(r) has the cdf F . The Algorithm 51 outlines the step for implementationof the inversion method.
B.2 Non-uniform Random Number Generator 219
Algorithm 51 Inversion methodInput: Inverse of the cdf F−1(x).Output: random number X with cdf F .1: generate a uniform random number r ∼ U(0,1)2: set X = F−1(r)3: return X
B.2.1.2 Rejection Method
The rejection method is an indirect method for generating random number X withpdf f . It uses an alternative proposal (or hat) distribution h(x) such that f (x) ≤ch(x) ∀x where c is constant and it is easier to generate a random number from hatfunction h than the original distribution f itself. The steps for the rejection methodare presented in Algorithm 52. The generation is composed of two steps. First, arandom number Y from the hat function h and a uniform random number r ∼ U(0,1)are generated. It then checks whether r ≤ f (Y )/ch(Y ) holds. If the test returns true,then the random number is accepted. Otherwise, the generation steps is repeated.
Algorithm 52 Rejection methodInput: a hat function h(x) such that f (x)≤ ch(x) where c is a constant.Output: a random number X with pdf f .1: repeat2: generate random number Y with hat pdf h3: generate a uniform random number r ∼ U(0,1)4: until (r ≤ f (Y )/ch(Y ))5: set X = Y and return X
B.2.1.3 Composition Method
The composition method assumes that pdf f can be decomposed as a finite mixture
f (x) =n
∑i=1
wi fi(x)
where fi’s are given density functions and wi’s are probability weights such thatwi > 0 for all i and ∑n
i=1 wi = 1. A random number X with pdf f can be obtainedby first generating a discrete random variate I with probabilities wi to decide whichpart fi should be chosen and then a random number with density fI is returned.Algorithm 53 outlines steps for an implementation of the composition method.
220 B Random Number Generation
Algorithm 53 Composition methodInput: decomposition of density f (x) = ∑n
i=1 wi fi(x) .Output: random number X with pdf f .1: generate random number I with probability vector wi
2: generate random number X with pdf fI
3: return X
B.2.2 Exponential Distribution
The exponential distribution Exp(λ ) is a continuous distribution with pdf definedas
f (x) =
{λe−λx , for x ≥ 0
0 , for x < 0
where λ > 0 is a parameter. The cdf F(x) of the distribution is thus given by
F(x) =
{1− e−λx , for x ≥ 0
0 , for x < 0.
A simple way to generate exponential random numbers is to apply the inversionmethod. It gives
F−1(r) =− 1λ
ln(1− r) =− 1λ
ln(r) (B.1)
where r ∼ U(0,1). The second equality is obtained by noting that if r is uniformlydistributed in (0,1), then so is (1−r). Algorithm 54 implements steps for generatingexponential random numbers.
Algorithm 54 Exponential random numberInput: rate parameter λOutput: random number X with exponential distribution Exp(λ ).1: generate a uniform random number r ∼ U(0,1)2: set X =− 1
λ ln(r)3: return X
B.2.3 Erlang Distribution
The Erlang distribution Erlang(k,λ ) is the distribution of the sum of k indepen-dent exponential variables with the same rate λ . The integer parameter k is calledshape parameter. The Erlang distribution has pdf
B.2 Non-uniform Random Number Generator 221
f (x) =λ kxk−1e−λx
(k−1)!, for x ≥ 0.
For the special case k = 1, the Erlang distribution Erlang(k,λ ) reduces to an ex-ponential distribution.
Algorithm 55 outlines the steps for generating an Erlang random number. It usesthe fact that the sum of k i.i.d exponential random numbers Xi ∼ Exp(λ ), i = 1 . . .k,with the same rate λ gives the Erlang random number.
Algorithm 55 Erlang random numberInput: integer shape parameter k, rate parameter λOutput: random number X with Erlang distribution Erlang(k,λ ).1: generate k exponential random numbers Xi ∼ Exp(λ )2: set X = ∑k
i=1 Xi
3: return X
B.2.4 Normal Distribution
Let N(0,1) be unit normal distribution with mean μ = 0 and variance σ2 = 1. It hasthe pdf
f (x) =1√2π
e−x2
2 .
A simple method for generating normal random numbers is the Box-Muller met-hod. It is based on the idea of generating the two-dimensional unit normal distribu-tion by generating a random angle and a random radius. Let (X ,Y ) be a pair of twoindependent unit normals. The joint pdf is a product
f (x,y) =1√2π
e−x2
21√2π
e−y2
2 =1
2πe−x2−y2
2 .
Consider the polar coordinate random variables (R,Θ) defined such that 0 ≤Θ ≤ 2π and X = Rcos(Θ) and Y = Rsin(Θ). It can be proved that Θ is uniformlydistributed in [0,2π] and R is a random variable with pdf
f (r) = re−r2
2 .
The angle Θ and the radius R, respectively, thus can be generated by directly ap-plying the conversion method. The generated values (Θ ,R) are then used to computethe pair (X ,Y ). The Box-Muller method is outlined in Algorithm 56.
222 B Random Number Generation
Algorithm 56 Unit normal random numberInput: unit normal distribution with mean 0 and variance 1Output: two random numbers X , Y with unit normal distribution N(0,1).1: generate two uniform random numbers r1,r2 ∼ U(0,1)2: compute θ = 2πr13: compute R =
√−2ln(r2)4: set X = Rcos(θ) and Y = Rcos(θ)5: return X and Y
B.2.5 Discrete Distribution with Given Probability Vector
Consider a discrete distribution where each outcome i is associated with a probabi-lity mass function (pmf) pi. If the support of the distribution is bounded from below,it is also called probability vector. The generation of discrete random number X gi-ven a probability vector (p1, p2, . . . , pm) is implemented in Algorithm 57. It is adirect application of the inversion method. Specifically, let F be the cdf of X andr ∼ U(0,1). The discrete random number is generated as
X = F−1(r) = min{i : F(i)≥ r}.
Algorithm 57 Discrete random numberInput: probability vector (p0, p1, . . . , pm).Output: a discrete random number X .1: generate uniform random number r ∼ U(0,1)2: set X = 0 and F = p03: while (r > F) do4: set X = X +1 and F = F + pX
5: end while6: return X
B.2.6 Poisson Distribution
The Poisson distribution Poi(λ ) is a discrete probability distribution that expressesthe probability of observing a number of events given its average rate λ . Let pk bethe probability that has k events occurring. It gives
pk =λ ke−λ
k!.
The generation of Poisson random numbers is implemented in Algorithm 58. Itis an application of the inversion method by using the fact that
B.2 Non-uniform Random Number Generator 223
pk+1 =λ
k+1pk.
Algorithm 58 Poisson random numberInput: rate parameter λ .Output: a Poisson random number X .1: generate uniform random number r ∼ U(0,1)2: set X = 0, p = e−λ and F = p3: while (r > F) do4: set X = X +15: compute p = λ p/X6: F = F + p7: end while8: return X
B.2.7 Binomial Distribution
The Binomial distribution Bin(n, p) is the discrete probability distribution of thenumber of successes in a sequence of n independent trials of which each has asuccess probability p. The probability pk of having exactly k successes in n trials is
pk =
(nk
)pk(1− p)n−k.
Algorithm 59 presents a simple generator for Binomial random numbers. Itcounts the number of successes in a series of n trials.
Algorithm 59 Binomial random numberInput: trials n and probability p.Output: a binomial random number X .1: set X = 02: for (i = 1 to n) do3: generate uniform random number r ∼ U(0,1)4: if (r ≤ p) then5: set X = X +16: end if7: end for8: return X
224 B Random Number Generation
B.2.8 Multinomial Distribution
The multinomial distribution Multi(n, p1, . . . , pM) is a multi-variate discrete dis-tribution. It generalizes the Binomial distribution where each trial can result in Moutcomes in which each outcome has a probability pk, for k = 1, . . .M.
The generation of a multinomial random vector is implemented in Algorithm 60.It is based on the fact that the number of successes of outcome k is a binomialrandom number.
Algorithm 60 Multinomial random numberInput: trials n and probability vector p1, p2, . . . pM .Output: multinomial random number.1: compute S = ∑M
i=1 pi
2: for (i = 1 to M) do3: generate a binomial random number Xi ∼ Bin(n, pi/S)4: set n = n−Xi
5: set S = S− pi
6: end for7: return Xis
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