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January Birthdays at a Call Center
Calls arrive at a call center according to a Poisson arrival process with an average rate of 2 calls/minute. Each caller has a 1/12 chance of having a January birthday, independent of other callers. What is the expected wait time until the call center receives 3 calls from callers with January birthdays?
Birds Arrive at a Bird Feeder
Birds arrive at a bird feeder according to a Poisson arrival process with a rate of 6 birds per hour. A person starts watching the feeder at time 0.
- What is the probability that the first three birds arrive at the feeder within 30 minutes?
- What is the expected time it takes for the 10th bird to arrive?
- 10% of the birds who visit the feeder are cardinals. What is the probability that the 3rd cardinal to arrive at the feeder is the 10th bird to arrive at the feeder?
- 10% of the birds who visit the feeder are cardinals. The person watching the feeder decides to continue watching the feeder until they see a cardinal. What is the probability that the person waits more than 5 hours?
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]
Poisson Thinning
Let \(N(t)\) be a Poisson process with intensity λ. For each occurrence, we flip a coin: if heads comes up we label the occurrence green, if tails comes up we label it red. The coin flips are independent and \(p\) is the probability to see heads.
- Show that the green occurrence form a Poisson process with intensity λp.
- Connect this with example 2.2.5 from Meester.
- We claim that the red occurrences on the one hand, and the green occurrences on the other hand form independent Poisson processes. Can you formulate this formally, and prove it , using Example 2.2.5 from Meester once more?
[Meester ex. 7.5.7]
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 \( \mathbf{E}\big(\,N(1,4)\,N(2,6)\,\big)\) ?
Joint arrival times
Let \(T_1\) and \(T_5\) be the times of the first and fifth arrival in a Poisson process with rate \(\lambda\). Find joint density of \(T_1\) and \(T_5\).
[Pitman p355 #12]
