Chapter 3 Flashcards
Define a counting process
A counting process is a continuous-time stochastic process
{Xt , t ∈ R+} with values in natural number sand zero such that
X0=0
Xt is number of events occuring over (0,t]
Xt is non decreasing and right continuous
Explain independent increments for a counting process
A counting process Xt , t ≥ 0 has independent increments if the
number of events which occur in disjoint time intervals are
independent, i.e. Xtn − Xtn−1 , . . . , Xt2 − Xt1
are independent for 0 ≤ t1 ≤ . . . ≤ tn
Define stationary increments
A counting process Xt , t ≥ 0 has stationary increments if the
number of events on intervals of the same length has the same
distributions
Define a poisson process - 4 checks
A counting process Xt , t ≥ 0 is a Poisson process with rate λ if
1 X0 = 0
2 Xt , t ≥ 0 has independent increments
3 Xt , t ≥ 0 has stationary increments
4 The number of events in any interval of length t is Poisson
distributed with a parameter λt,
Explain what the rate outlines
In one time unti we expect lamda events
Define arrival times
Random variable Tk = inf{t ≥ 0 : Xt = k} (and by convention T0 = 0) are said to be arrival times
Define interarrival times
For any k ≥ 1, the random variables τk = Tk − Tk−1 are said to be
interarrival times.
How are the interarrival times distributed
The set of τk , k > 1 are i.i.d. random variables with the
exponential distribution with the parameter λ
What distribution that is continuous has the memoryless property
Exponential distirbutions
What is the distribution of the arrival times
Erlang distribution - special case of gamma distribution where the parameter k is a positive integer
How is T1 given Xt=1 distributed over the interval (0,t]
Unfiormly
What does it mean to consider the order statistics?
Considering: the order statistics Y(1) < · · · < Y(n) - random variables defined by sorting the values of Y1, . . . , Yn in increasing order.
Define a marked poisson process
A marked Poisson process is:
a Poisson process {Xt , t ≥ 0} with arrival times T1, T2, …
a collection of iid random variables M1, M2, … indep. of {Xt }
Give an example of Tk and Mk for a marked poisson process
Tk is the arrival time of the k-th customer.
Mk is the money spent by the k-th customer
In words how do we find the thinned poisson process?
In other words, {Yt } is obtained from {Xt } by erasing the arrivals
Ti with Mi = 0.
