Question 1

What is the Poisson distribution used for?

Accepted Answer

The Poisson distribution models counts of independent rare events over a fixed interval. Classic examples: customer calls per hour at a call centre; page requests per second on a web server; typos per page in a manuscript; traffic accidents per week at an intersection; software bugs per 1,000 lines of code; radioactive decay counts per minute. The key requirement: events occur independently at a stable average rate.

Question 2

What are the mean and variance of a Poisson distribution?

Accepted Answer

Both the mean and variance equal λ: μ = λ and σ² = λ, so SD = √λ. This equality is a unique property of the Poisson distribution. If the sample variance is much larger than the mean (overdispersion), a negative binomial model may fit better. If variance << mean (underdispersion), consider a different model or verify the independence assumption.

Question 3

What are the conditions for a Poisson process?

Accepted Answer

1) Independence: one event does not affect another. 2) Stationarity: constant average rate λ across the interval. 3) Ordinarity: two events cannot occur at exactly the same instant. 4) Proportionality: probability of an event in a short sub-interval is proportional to its length. When all four hold, event counts follow a Poisson distribution and waiting times follow an exponential distribution.

Question 4

When should I use Poisson vs binomial?

Accepted Answer

Use Poisson when: counting events in a continuous interval (time/space), there is no fixed upper limit on the count, and the average rate is stable. Use binomial when: there is a fixed number of n trials each with probability p. Rule of thumb: if n > 20 and p < 0.05 (or np < 10), use Poisson with λ = np as a good approximation. For n=500 and p=0.002 (λ=1), Poisson is essentially exact.

Question 5

How do I interpret a Poisson probability?

Accepted Answer

P(X=k) is the probability of exactly k events in one interval given average rate λ. Example: server receives λ=3 requests/second. P(X=5) = e⁻³ × 3⁵ / 5! = 0.0498 × 243 / 120 ≈ 0.101. There's a 10.1% chance of exactly 5 requests in the next second. P(X≤5) ≈ 0.916 means a 91.6% chance of at most 5 requests. P(X>5) ≈ 0.084 means 8.4% chance of 6 or more.

Question 6

What is the relationship between Poisson and exponential distributions?

Accepted Answer

If events follow a Poisson process with rate λ (events per unit time), then the waiting time between events follows an exponential distribution with mean 1/λ. Example: if a call centre receives λ=4 calls/minute, the wait between consecutive calls is Exponential with mean 15 seconds. This Poisson–exponential duality is the foundation of queuing theory (M/M/1 queues and variants).

Domain	Event	Typical λ
Call centre	Calls per minute	3–8
Web server	Requests per second	10–100
Insurance	Claims per week	2–15
Hospital A&E	Admissions per hour	4–20
Software QA	Bugs per 1,000 LOC	1–5
Nuclear physics	Decay events per minute	0.1–50

Criterion	Poisson	Binomial
Number of trials	No fixed upper limit	Fixed n
Rate type	Events per interval (time/area)	Probability per trial
Parameter	λ (rate)	n and p
Mean	λ	np
Mean = Variance?	Yes (always)	Only if p = 0.5
Best when	n large, p small, λ = np moderate	n moderate, p not extreme

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