Non return valves water regulations

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LECTURE 13The Bernoulli processA sequence of independentThe Bernoulli process•Bernoulli trials•Readings:Section 6.1•At each trial,i:–P(success) =P(Xi= 1) =pLecture outline–P(failure) =P(Xi= 0) = 1−p•Definition of Bernoulli processExamples:•Random processes•–Sequence of lottery wins/losses•Basic properties of Bernoulli process–Sequence of ups and downs of the Dow•Distribution of interarrival timesJones–Arrivals (each second) to a bank•The time of thekth success–Arrivals (at each time slot) to server•Merging and splittingRandom processesNumber of successesSinntime slots•First view:•P(S=k)=sequence of random variablesX1,X2,...•E[Xt]=•E[S]=•Var(Xt)=•Var(S)=•Second view:what is the right sample space?•P(Xt=1forallt)=•Random processes we will study:–Bernoulli process(memoryless, discrete time)–Poisson process(memoryless, continuous time)–Markov chains(with memory/dependence across time)1
Interarrival timesTime of thekth arrival•T1: number of trials until first success•Given that first arrival was at timeti.e.,T1=t:–P(T1=t)=additional time,T2, until next arrival–Memoryless property–has the same (geometric) distribution–independent ofT1–E[T1]=–Var(T1)=•Yk: number of trials tokth success–E[Y•If you buy a lottery ticket every day, whatk]=is the distribution of the length of thefirst string of losing days?–Var(Yk)=–P(Yk=t)=Splitting of a Bernoulli ProcessMerging of Indep. Bernoulli Processes(using independent coin flips)Bernoulli (p)timetimeMerged process:Bernoulli (pqpqq+−)timeOriginalprocesstimeBernoulli (q)1−qtimetimeyields a Bernoulli processyields Bernoulli processes(collisions are counted as one arrival)Sec. 6.1The Bernoulli Process305Splitting and Merging of Bernoulli ProcessesStarting with a Bernoulli process in which there is a probabilitypof an arrivalat each time, considersplittingit as follows. Whenever there is an arrival, wechoose to either keep it (with probabilityq), or to discard it (with probability1−q); see Fig. 6.3. Assume that the decisions to keep or discard are independentfor different arrivals. If we focus on the process of arrivals that are kept, we seethat it is a Bernoulli process: in each time slot, there is a probabilitypqof akept arrival, independent of what happens in other slots. For the same reason,the process of discarded arrivals is also a Bernoulli process, with a probabilityof a discarded arrival at each time slot equal top(1−q).Figure 6.3:Splitting of a Bernoulli process.In a reverse situation, we start with twoindependentBernoulli processes(with parameterspandq, respectively) andmergethem into a single process,as follows. An arrival is recorded in the merged process if and only if thereis an arrival in at least one of the two original processes. This happens withprobabilityp+q−pq[one minus the probability (1−p)(1−q) of no arrival ineither process]. Since different time slots in either of the original processes areindependent, different slots in the merged process are also independent. Thus,the merged process is Bernoulli, with success probabilityp+q−pqat each timestep; see Fig. 6.4.Splitting and merging of Bernoulli (or other) arrival processes arises inmany contexts. For example, a two-machine work center may see a stream ofarriving parts to be processed and split them by sending each part to a randomlychosen machine. Conversely, a machine may be faced with arrivals of differenttypes that can be merged into a single arrival stream.The Poisson Approximation to the BinomialThe number of successes innindependent Bernoulli trials is a binomial randomvariable with parametersnandp, and its mean isnp. In this subsection, we306The Bernoulli and Poisson ProcessesChap. 6Figure 6.4:Merging of independent Bernoulli processes.concentrate on the special case wherenis large butpis small, so that the meannphas a moderate value. A situation of this type arises when one passes fromdiscrete to continuous time, a theme to be picked up in the next section. Forsome examples, think of the number of airplane accidents on any given day:there is a large numbernof trials (airplane flights), but each one has a verysmall probabilitypof being involved in an accident. Or think of counting thenumber of typos in a book: there is a large number of words, but a very smallprobability of misspelling any single one.Mathematically, we can address situations of this kind, by lettingngrowwhile simultaneously decreasingp, in a manner that keeps the productnpat aconstant valueλ. In the limit, it turns out that the formula for the binomial PMFsimplifies to the Poisson PMF. A precise statement is provided next, togetherwith a reminder of some of the properties of the Poisson PMF that were derivedin Chapter 2.Poisson Approximation to the Binomial•A Poisson random variableZwith parameterλtakes nonnegativeinteger values and is described by the PMFpZ(k)=e−λλkk!,k=0,1,2,....Its mean and variance are given byE[Z]=λ,var(Z)=λ.2