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Conditioning a Poisson Arrival Process
Consider a Poisson process with parameter \(\lambda\). What is the conditional probability that \(N(1) = n\) given that \(N(3) = n\)? (Here, \(N(t) \) is the number of calls which arrive between time 0 and time \(t\). ) Do you understand why this probability does not depend on \(\lambda\)?
[Meester ex 7.5.4]
Covariance of a Branching Process
Show that for a branching process \( (Z_n)\) with expected offspring \(\mu\) one has
\[\mathbf{E}( Z_n Z_m)= \mu^{n-m} \mathbf{E}( Z_m^2)\]
for \(0\leq m\leq n\).
Basic Branching Process
Consider the branching process with offspring distribution given by \( P(X=0)=\alpha\), \( P(X=1)=\frac23-\alpha\) and \( P(X=2)=\frac13\) for some \(\alpha \in [0,\frac23]\).
- For what values of \(\alpha\) is the process certain to die out. ?
- For values where there is a probability of surviving forever, what is this probability as a function of \(\alpha\) ?
Geometric Branching Process
Consider a branching process with a geometric offspring distribution \( P(X=k) = (1-p)p^k\), for \(k=0,1,2,\dots\) . Show that the ultimate extinction is certain if \(p \leq \frac12\) and that the probability of extinction is \((1-p)/p \) if \(p > \frac12\).
[Meester ex. 6.6.5]
Closest Point
Consider a Poisson random scatter of points in a plane with mean intensity \(\lambda\) per unit area. Let \(R\) be the distance from zero to the closest point of the scatter.
- Find a formula for the c.d.f. and the density of \(R\) and sketch their graphs.
- Show that \(\sqrt{2 \lambda \pi} R\) has the Rayleigh distribution.
- Find the mean and mode of \(R\).
[pitman p 389, # 21]
Joint Density of Arrival Times
Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?
Calls arriving
Assume that calls arrive at a call centre according to a Poisson arrival process with a rate of 15 calls per hour. For \(0 \leq s < t\), let \(N(s,t)\) denote the number of calls which arrive between time \(s\) and \(t\) where time is measured in hours.
- What is \( \mathbf{E}\big(\,N(3,5)\,\big)\) ?
- What is the second moment of \(N(2,4) \) ?
- What is \( \mathbf{E}\big(\,N(1,4)\,N(2,6)\,\big)\) ?
Memorylessness and the Geometric Distribution
Let \(X\) be a random variable with range \(\{0,1,2,3,\dots\}\) and distributed geometrical with probability \(p\).
- Show that for every \(n \geq 0\) and \(j\geq 0\),
\[ \mathbf{P}(X-n=j \mid X\geq n) = \mathbf{P}(X=j)\] - If \(X \) is the time to the failure of a machine, then \(\mathbf{P}( X\geq n) \) is the event that the machine has not failed by time \(n\). Why is the above property called Memorylessness ?
- Show that the geometric distribution is the only random variable with range equal to \(\{0,1,2,3,\dots\}\) with this property.
