Superposition. Thinning

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1 Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2, calld th suprposition of th procsss, is again a Poisson procss but with rat λ 1 + λ 2. Th proof is straight forward from Dfinition 5.3 and hnc omittd. Rmark: By rpatd application of th abov argumnts w can s that th suprposition of k indpndnt Poisson procsss with rats λ 1,, λ n is again a Poisson procss with rat λ λ n. STAT253/ Wintr Lctur 11-1 STAT253/ Wintr Lctur 11-2 Why Poisson Procsss Mak Sns? Thr is a usful rsult in probability thory which says that: Hr if w tak N indpndnt counting procsss and sum thm up, thn th rsulting suprposition procss is approximatly a Poisson procss. N must b larg nough and th rats of th individual procsss must b small rlativ to N but th individual procsss that go into th suprposition can othrwis b arbitrary. Thinning Considr a Poisson procss with rat λ. At ach arrival of vnts, it is classifid as a { Typ 1 vnt with probability p or Typ 2 vnt with probability 1 p, indpndntly of all othr vnts. Lt N i (t) # of typ i vnts occurrd during [, t], i 1, 2. Not that N(t) N 1 (t) + N 2 (t). Proposition 5.2 {N 1 (t), t } and {N 2 (t), t } ar both Poisson procsss having rspctiv rats λp and λ(1 p). Furthrmor, th two procsss ar indpndnt STAT253/ Wintr Lctur 11-3 STAT253/ Wintr Lctur 11-4

2 Proof of Proposition 5.2 First obsrv that givn N(t) n + m, (N 1 (t), N 2 (t)) Binomial(n + m, p, (1 p)). (why?) Thus P(N 1 (t) n, N 2 (t) m) P(N 1 (t) n, N 2 (t) m N(t) n + m)p(n(t) n + m) ( n + m )p n (1 p) m λt (λt)n+m n (n + m)! λtp (λpt)n λt(1 p) (λ(1 p)t)m n! m! P(N 1 (t) n)p(n 2 (t) m). This provs th indpndnc of N 1 (t) and N 2 (t) and that N 1 (t) Poisson(λpt), N 2 (t) Poisson(λ(1 p)t). Both {N 1 (t)} and {N 2 (t)} inhrit th stationary and indpndnt incrmnt proprtis from {N(t)}, and hnc ar both Poisson procsss. STAT253/ Wintr Lctur 11-5 Som Convrs of Thinning & Suprposition (Cont d) Lt N(t) N A (t) + N B (t) b th suprposition of th two procsss. Lt I i { 1 if th ith vnt in th suprpositon procss is an A vnt othrwis Th I i, i 1, 2,... ar i.i.d. Brnoulli(p). whr p Approach 2: λ A λ A + λ B. P(S A n < S B 1 ) P(th first n vnts ar all A vnts) ( λa λ A + λ B P(S A n < S B m) P(at last n A vnts occur bfor m B vnts) ) n P(gtting at last n hads bfor m tails) n+m 1 ( ) ( ) n + m 1 k ( λa λb k λ A + λ B λ A + λ B kn. ) n+m 1+k STAT253/ Wintr Lctur 11-7 Som Convrs of Thinning & Suprposition Considr two indp. Poisson procsss {N A (t)} and {N B (t)} w/ rspctiv rats λ A and λ B. Lt Find P(S A n < S B m). S A n arrival tim of th mth A vnt S B m arrival tim of th nth B vnt Approach 1: Obsrvr that S A n Gamma(m, λ A ), S B m Gamma(n, λ B ) and thy ar indpndnt. Thus P(Sn A < Sm) B x<y λ A λ Ax (λ Ax) n 1 (n 1)! λ B λ By (λ By) m 1 (m 1)! dxdy STAT253/ Wintr Lctur 11-6 Proposition 5.3 (Gnralization of Prop. 5.2) Considr a Poisson procss with rat λ. If an vnt occurs at tim t will b classifid as a typ i vnt with probability p i (t), i 1,..., k, i p i(t) 1, for all t, indpndntly of all othr vnts. thn N i (t) numbr of typ i vnts occurring in [, t], i 1,..., k. Not N(t) k N i(t). Thn N i (t), i 1,..., k ar indpndnt Poisson random variabls with mans λ p i(s)ps STAT253/ Wintr Lctur 11-8

3 Exampl (Rvision of Exrcis 6.66 on p.364) Policyholdrs of a crtain insuranc company hav accidnts occurring according to a Poisson procss with rat λ. Th amount of tim T from whn th accidnt occurs until a claim is mad has distribution G(t) P(T t). Lt N c (t) b th numbr of claims mad by tim t. Find th distribution of N c (t). Solution. Considr an accidnt occurrd at tim s. Thn it is claimd by tim t if s + T t, i.., with probability p(s) P(T t s) G(t s). By Proposition 5.3, N c (t) has a Poisson distribution with man λ p(s)ps λ G(t s)ds λ G(s)ds STAT253/ Wintr Lctur 11-9 Proposition 5.4 Lt {N 1 (t), t }, and {N 2 (t), t } b two indpndnt nonhomognous Poisson procss with rspctiv intnsity functions λ 1 (t) and λ 2 (t), and lt N(t) N 1 (t) + N 2 (t). Thn (a) {N(t), t } is a nonhomognous Poisson procss with intnsity function λ 1 (t) + λ 2 (t). (b) Givn that an vnt of th {N(t), t } procss occurs at tim t thn, indpndnt of what occurrd prior to t, th vnt at t was from th {N 1 (t)} procss with probability λ 1 (t) λ 1 (t) + λ 2 (t). STAT253/ Wintr Lctur Nonhomognous Poisson Procss Dfinition 5.4a. A nonhomognous (a.k.a. non-stationary) Poisson procss with intnsity function λ(t) is a counting procss {N(t), t } satisfying (i) N(). (ii) having indpndnt incrmnts. (iii) P(N(t + h) N(t) 1) λ(t)h + o(h). (iv) P(N(t + h) N(t) 2) o(h). Dfinition 5.4b. A nonhomognous Poisson procss with intnsity function λ(t) is a counting procss {N(t), t } satisfying (i) N(), (ii) for s, t, N(t + s) N(s) is indpndnt of N(s) (indpndnt incrmnt) (iii) For s, t, N(t + s) N(s) Poisson(m(t + s) m(s)), whr m(t) λ(t)dt Th two dfinitions ar quivalnt STAT253/ Wintr Lctur Compound Poisson Procsss Dfinition. Lt {N(t)} b a (homognous) Poisson procss with rat λ and Y 1, Y 2,... ar i.i.d random variabls indpndnt of {N(t)}. Th procss X (t) N(t) Y i is calld a compound Poisson procss, in which X (t) is dfind as if N(t). A compound Poisson procss has indpndnt incrmnt, sinc X (t + s) X (s) N(t+s) N(s) Y i+n(s) is indpndnt of X (s) N(s) Y i, and stationary incrmnt, sinc X (t + s) X (s) N(t+s) N(s) Y i+n(s) has th sam distribution as X (t) N(t) Y i STAT253/ Wintr Lctur 11-12

4 Th Man of a Compound Poisson Procss Suppos E[Y i ] µ Y, Var(Y i ) σy 2. Not that E[N(t)] λt. E[X (t) N(t)] N(t) E[Y i N(t)] N(t) E[Y i] (sinc Y i s ar indp. of N(t)) N(t)µ Y Thus E[X (t)] E[E[X (t) N(t)]] E[N(t)]µ Y λtµ Y Varianc of a Compound Poisson Procss (Cont d) Similarly, using that E[N(t)] Var(N(t)) λt, w hav ( N(t) ) Var[X (t) N(t)] Var Y i N(t) N(t) Var(Y i N(t)) N(t) Var[Y i] (sinc Y i s ar indp. of N(t)) N(t)σ 2 Y E[V [X (t) N(t)]] E[N(t)]σ 2 Y λtσ2 Y Var(E[X (t) N(t)]) Var(N(t)µ Y ) Var(N(t))µ 2 Y λtµ2 Y Thus Var(X (t)) E[V [X (t) N(t)]] + Var(E[X (t) N(t)]) λt(σ 2 Y + µ2 Y ) λte[y 2 i ] STAT253/ Wintr Lctur STAT253/ Wintr Lctur CLT of a Compound Poisson Procss Conditional Poisson Procsss As t, th distribution of X (t) E[X (t)] X (t) λtµ Y Var(X (t)) λt(σy 2 + µ2 Y ) convrgs to a standard normal distribution N(, 1). Dfinition. A conditional (or mixd) Poisson procss {N(t), t } is a counting procss satisfying (i) N(), (ii) having stationary incrmnt, and (iii) thr is a random variabl Λ > with probability dnsity g(λ), such that N(t + s) N(s) Λ Poisson(λt), i.., P(N(t + s) N(s) k) g(λ)dλ, k, 1,... STAT253/ Wintr Lctur STAT253/ Wintr Lctur 11-16

5 Rmark: In gnral, a conditional Poisson procss dos NOT hav indpndnt incrmnt. P(N(s) j, N(t + s) N(s) k) ( λs (λs)j j! λs (λs)j g(λ)dλ j! g(λ)dλ ) ( P(N(s) j)p(n(t + s) N(s) k) ) g(λ)dλ STAT253/ Wintr Lctur 11-17