# Category Archives: Poisson arrivial process

## 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.

1. What is the probability that the first three birds arrive at the feeder within 30 minutes?
2. What is the expected time it takes for the 10th bird to arrive?
3. 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?
4. 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.

1.  Show that the green occurrence form a Poisson process with intensity λp.
2. Connect this with example 2.2.5 from Meester.
3. 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.

1. What is $$\mathbf{E}\big(\,N(3,5)\,\big)$$ ?
2. What is the second moment of $$N(2,4)$$ ?
3. 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]