Question

3
.
Let Nt be a (homogeneous) Poisson process with intensity lambda. Find the limit limh->0 P{Nh = 0}:

1)1
2) 1 – lambda.h
3) lambda.h
4) 0

note from the asker:
Nt: 't' is subscript
Nh: 'h' is subscript
h->0 is under lim​

Answer 1

Answer:

We present here the essentials of the Poisson point process with its many interesting

properties. As preliminaries, we first define what a point process is, define the renewal

point process and state and prove the Elementary Renewal Theorem.

1.1 Point Processes

Definition 1.1 A simple point process ψ = {tn : n ≥ 1} is a sequence of strictly increasing points

0 < t1 < t2 < · · · , (1)

with tn−→∞ as n−→∞. With N(0) def = 0 we let N(t) denote the number of points that

fall in the interval (0, t]; N(t) = max{n : tn ≤ t}. {N(t) : t ≥ 0} is called the counting

process for ψ. If the tn are random variables then ψ is called a random point process.

We sometimes allow a point t0 at the origin and define t0

def = 0. Xn = tn − tn−1, n ≥ 1,

is called the n

th interarrival time.

We view t as time and view tn as the n

th arrival time (although there are other kinds of

applications in which the points tn denote locations in space as opposed to time). The

word simple refers to the fact that we are not allowing more than one arrival to ocurr at

the same time (as is stated precisely in (1)). In many applications there is a “system” to

which customers are arriving over time (classroom, bank, hospital, supermarket, airport,

etc.), and {tn} denotes the arrival times of these customers to the system. But {tn} could

also represent the times at which phone calls are received by a given phone, the times

at which jobs are sent to a printer in a computer network, the times at which a claim is

made against an insurance company, the times at which one receives or sends email, the

times at which one sells or buys stock, the times at which a given web site receives hits,

or the times at which subways arrive to a station. Note that

tn = X1 + · · · + Xn, n ≥ 1,

the n

th arrival time is the sum of the first n interarrival times.

Also note that the event {N(t) = 0} can be equivalently represented by the event

{t1 > t}, and more generally

{N(t) = n} = {tn ≤ t, tn+1 > t}, n ≥ 1.

In particular, for a random point process, P(N(t) = 0) = P(t1 > t).

Step-by-step explanation:

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