Karteikarten
Probability: Definitions
Karten in diesem Stapel (26)
- 1Poisson Process with rate LZn~exp(L), iid
Tn=Z0+...+Z(n-1)
Xt=#{n>0: Tn - 2Poisson Markov Property-P(X(t+s)=L|Xt=k,....)=P(X(t+s)=L|Xt=k)=Pk(Xs=L)
-others? - 3X, Y Conditionally Independent given ZP(X=A, Y=B|Z=z)=P(X=A|Z=z)P(Y=B|Z=z)
- 4Simple Birth Process of Rates (Ln) -(k, (Ln))-birth processXt=k+#{n>0: Tn
- 5ExplosionP(Tinf < inf)>0
Tinf=limTn=SUM Zj
infinitely many births in finite time - 6Birth Process Markov PropertyX a (k, (Ln)) birth process , m>k
(Xr)r0 conditionally independent given Xt=m
(X(t+s))s>0 a (m, (Ln)) birth process - 7Strong Markovl
- 8Continuous Markov Chain with holding rates (Li)Mn a markov chain
Zn such that, conditionally given M0=i0,...,Mn=in, we have Zj~(Lj), independent up to j=n
T0=0, Tn=Z0+...+Z(n-1)
Xt=Mn if Tn - 9Q-Matrixv initial distribution, Hi,j jump probabilities, Li holding rates
satisfying Hi,i=0 or =1 and Li=0 iff Hi,i=1
Define Q=(qi,j) st qi,j=LiHi,j for not j=i
qi,i=-Li - 10i communicates with jPi(Xt=j for some t>0)>0
starting from i, there is a possibility of reaching j - 11Closed ClassNo i in the class and j not in the class s.t i communicates with j
- 12Absorbing State{i} closed
- 13IrreducibleOne communicating class
- 14RecurrentPi({t>0:Xt=i} is unbounded)=1
- return to state i an infinite number of times - 15Positive RecurrentLi=0 or
mi=Ei(Hi) - 16Null recurrentexpected time to return to state i is infinite
- 17TransientPi({t>0:Xt=i} is unbounded)=0
- 18Invariant Distribution&P(t)=& for all t>0
- 19Xn converges to X almost surelyP(Xn converges to X)=1
ie for all e>0, there exists an N s.t for all n>N |Xn-X| - 20Xn converges to X in probabilityfor all e>0, P(|Xn-X|>e) converges to 0
- 21Xn converges to X in distributionfor all x s.t it is continuous, P(Xn
- 22Xn converges to X in L1E(|Xn|)
- 23Renewal ProcessZn iid positive random variables
Tn=Z0+...+Z(n-1)
Xt=#{n>0:Tn - 24(Undelayed) Renewal Functionm(t)=E(Xt)
- 25Delayed Renewal Process and Delay DistributionZn>0 iid and Z0 independent and different
Xt=#{n>0: Tn - 26Size-Biased Probability Function-probability function P, m=SUMnPn, Pn(sb)=(nPn)/m
-probability density function f, m=integral(tf(t))dt, f(sb)(z)=(zf(z))/m
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- Beschreibung
- Mathematics Karteikarten am Probability: Definitions, erstellt von Joanna Engle am 05/04/2019.
- Erstellt von
- Joanna Engle
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