Comprehension check — comprehensive review and full-question audit
On a fresh case separate from the running example, sweep through interval scaling, the pmf, cumulative probability, the mode, the sum of Poissons, relative spread, simulator readouts, and minimal-implementation reading in one pass.
Run the full flow on a fresh case
Here we check whether every formula so far carries over intact to a different case. We revisit interval scaling, the pmf, cumulative probability, the mode, and the sum of Poissons — all in one pass.
The course at a glance: introduction → pmf → mean and variance → interval scaling → implementation.
The case for the comprehensive review
Suppose chat inquiries arrive at an average of 3 per 30 minutes. We will also consider a separate monitoring-alert stream that averages 2 per 30 minutes.
| Item | Content |
|---|---|
| Inquiries | Average of 3 per 30 minutes |
| Monitoring alerts | Average of 2 per 30 minutes (independent) |
| Approximation | e^-3 ≈ 0.0498 |
Comprehensive review — run the full flow on a fresh case
Suppose chat inquiries arrive at an average of 3 per 30 minutes. Suppose also a separate monitoring-alert stream averages 2 per 30 minutes (independent).
Feel free to use e^-3 ≈ 0.0498.
Q1. For the inquiry stream (3 per 30 minutes on average), what is λ for a 10-minute window?
3 × (10 / 30) = 1.
Q2. For the inquiry stream (3 per 30 minutes on average), what is the probability P(X = 0) of exactly 0 events in 30 minutes?
P(X = 0) = e^-3 ≈ 0.050.
Q3. Under the same conditions, what is P(X = 2)?
e^-3 × 3^2 / 2! ≈ 0.224.
Q4. Under the same conditions, what is P(X ≤ 1)?
P(X ≤ 1) = P(0) + P(1) = e^-3 + 3e^-3 ≈ 0.199.
Q5. For the inquiry stream (3 per 30 minutes on average), which is the most likely count?
λ = 3 is an integer, so the mode candidates tie at 2 events and 3 events.
Q6. Combining inquiries (average 3) and monitoring alerts (average 2), what is λ for the total count over 30 minutes?
Independent Poissons add their means: 3 + 2 = 5.
Q7. Comparing a system with an average of 3 per 30 minutes against one with an average of 12 per 2 hours, which has the smaller relative spread (coefficient of variation) in the sense of std. dev. / mean?
Relative spread (coefficient of variation) is 1 / √λ, so λ = 12 gives the smaller value.
Comprehensive review — simulator readouts and minimal implementation
Confirm that the Chapter 5 simulator and the Chapter 6 minimal implementation are consistent with the formulas you have been computing by hand.
Q1. When you select the Very busy preset (λ = 9) in the Chapter 5 simulator, which value should the relative spread (coefficient of variation, std. dev. / mean) display?
1 / √9 = 1 / 3 ≈ 0.333. The simulator readout shows the same value.
Q2. How should the loop condition while (product > limit) in samplePoissonKnuth from Chapter 6 be interpreted?
Since limit = e^(−λ), the inequality U_1 × ... × U_n > e^(−λ) is equivalent to E_1 + ... + E_n < 1, where E_i = -ln(U_i) / λ. The loop runs while events keep arriving inside the unit interval and stops the moment the cumulative gap exceeds 1.
Where this course leaves you
- When you see count data over a fixed window, reaching for Poisson becomes a natural instinct.
- You can compute
P(X = k)andP(X ≤ k)by hand for small λ. - You can explain mean = variance = λ, interval scaling, and the sum of Poissons both in words and in formulas.
- You can read the minimal vanilla JavaScript implementation and drop it into a small calculator of your own.