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]
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Posted in Poisson arrivial process
Tagged JCM_math230_HW8_F22, JCM_math230_HW8_S15, JCM_math340_HW9_F13
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 \(0\leq m\leq n\).
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Posted in Branching Processes
Tagged JCM_math340_HW9_F13
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 \(\alpha \in [0,\frac23]\).
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Posted in Basic Stochastic Processes, Branching Processes
Tagged JCM_math340_HW9_F13
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]
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Posted in Basic Stochastic Processes, Branching Processes, Exponential Random Variables
Tagged JCM_math340_HW9_F13
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.
[pitman p 389, # 21]
Posted in Poisson
Tagged JCM_math230_HW8_S13, JCM_math230_HW9_F22, JCM_math230_HW9_S15, JCM_math340_HW9_F13
Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?
Posted in Poisson
Tagged JCM_math230_HW10_S15, JCM_math230_HW8_S13, JCM_math340_HW9_F13
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.
Let \(X\) be a random variable with range \(\{0,1,2,3,\dots\}\) and distributed geometrical with probability \(p\).
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Posted in Geometric Distribution
Tagged JCM_math340_HW9_F13
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