Flashcards

SB3.1

Ben Tidswell Part B uni Inglês 51 flashcards

Let X=(X_t)_t≥0 be a Poisson process of rate λ started from 0. State the Markov property

Flashcards do conjunto (51)

1Let X=(X_t)_t≥0 be a Poisson process of rate λ started from 0. State the Markov property
2Let X=(X_t)_t≥0 and Y=(Y_t)_t≥0 be independent Poisson processes of rates λ and μ respectively. State the Superposition of Poisson processes theorem
3Let Z be a Poisson process of rate λ and let p∈[0,1]. State the Thinning of a Poisson process theorem
4Let X~Markov(ν,Q) and T be a stopping time. State the Strong Markov property
5Let X=(X_t)_t≥0~Markov(ν,Q) be a minimal positive recurrent irreducible continuous time Markov chain and let ξ be the invariant distribution. State the Ergodic theorem
6Let X be a renewal process with 𝔼[Z_1]=μ∈(0,∞). State the Strong law of renewal theory.
7Let X=(X_t)_t≥0 be a renewal process whose inter-arrival times Z_1,Z_2,... satisfy 𝔼[Z_1]=μ∈(0,∞) and 0<σ^2=Var(Z_1)<∞. State the Central limit theorem of renewal theory.
8Let X be a renewal process with mean inter-arrival time μ and m(t)=𝔼[X_t]. State The elementary renewal theorem.
9Let X be a (possibly delayed) renewal process having continuous inter-arrival distribution of finite mean μ and let m(t)=𝔼[X_t]. State the Renewal theorem.
10Let X be a renewal process with continuous inter-arrival distribution of mean μ<∞. State the Key renewal theorem
11Write p_ij(s)=ℙ[X_(t+s)=j|X_t=i] and P(s)=(p_ij(s))_(i,j∈𝕊) as a matrix. Prove that (P(t))_(t≥0) is a semigroup
12Suppose that X~Markov(ν,Q). State and prove (not uniqueness) the backward equation theorem
13Define a simple birth process
14Define a Q-matrix
15Define a renewal process
16Define a delayed renewal process
17State what it means for a process X to have stationary increments
18State the mean of a Beta(a,b) distribution
19State the renewal property
20Suppose (p_n)_(n≥0) is a p.m.f. on ℕ with mean μ_1<∞ and f is a p.d.f. on [0,∞) with mean μ_2<∞. State the size biased distributions of p and f
21Let X be a delayed renewal process with inter-arrival times Z_1,Z_2,.... State a theorem relating a special case of the distribution of Z_1 to the distribution of the excess life
22Suppose that F is a distribution on [0,∞) with density f and mean μ<∞. Then if L has the size-biased distribution and U~Uniform[0,1] independently of L, state ℙ[LU>x] and the density g(x) of LU
23State the Jump-chain/holding time definition of a minimal continuous time Markov chain
24What does it mean to say that explosion is possible for a simple birth process X with rates (λ_n)_n≥0
25Suppose that X~Markov(ν,Q). State 3 conditions such that, if any of them hold, X does not explode
26State the infinite sum of a geometric series with first term a and constant ratio r
27State the mean and variance of an exponential distribution
28Let X be a CTMC. Define what it means for a state to be recurrent/transient
29Let X be a CTMC. Define what it means for a recurrent state to be positive recurrent/null recurrent
30Let X be a CTMC with Q-matrix Q. State the detailed balance equations for distributions ξ
31Define the M/M/1 queue
32Let X be a CTMC with Q-matrix Q. State an if and only if condition for X to have an invariant distribution ξ
33State the sum of the first n terms of a geometric series with first term a and constant ratio r
34State the conditional probability formula
35Let I be a finite or countably infinite index-set. Let {E_i:i∈I} be independent random variables such that E_i~Exp(λ_i) for i∈I, where λ_i≥0 and Σ_(i∈I)λ_i<∞. State the distribution of M:=inf_(i∈I)E_i and ℙ[E_k $Let I be a finite or countably infinite index-set. Let {E_i:i∈I} be independent random variables such that E_i~Exp(λ_i) for i∈I, where λ_...$
36Define what it means for a CTMC X to be minimal
37Let X be a CTMC with Q-matrix Q and transition matrices P(t), define what it means for a measure ξ to be an invariant distribution and give an if and only if statement regarding the Q-matrix under certain conditions
38Let X_t be a CTMC, define the jump chain for X
39State a necessary condition for explosion to occur with probability 1 (or 0) for a simple birth process started at k
40Let Y_t be a Yule process of rate λ, prove that m(t)=e^(λt), for t≥0
41Define the traffic intensity of an M/M/1 queue and state it's relation to recurrence/transience
42Let X be a CTMC. Define the time-reversal of X
43Consider an M/M/1 queue in equilibrium. State Burke's theorem
44Define the convolution product f*g and state what it represents in terms of probability distributionsboth integrals should be -inf to inf in general, 0 to t for strictly positive
45Let X be a renewal process with inter-arrival density f. Prove that m(t)=Σ_(k=1)^(∞)F_k(t)=∫_(0)^(t)Σ_(k=1)^(∞)f^(*(k))(s)ds
46Let X be a renewal process with inter-arrival density f. Show that m(t) satisfies the renewal equation m(t)=F(t)+∫_(0)^(t)m(t-s)f(s)ds
47State the mean of a geometric(p) distribution (trial number of first success)
48State the distribution of the sum of n Exp(λ) random variables
49Show that if X is a Poisson process of rate λ then X_t~Po(λt) for all t≥0
50Let X be a CTMC. Define a communicating class of 𝕊
51Let X be a minimal CTMC with jump chain Y. For i,j∈𝕊 state 4 equivalent statements to i→j