Chapter 5

Simulator — explore PMFs interactively

Drive λ from the simulator and cross-check the peak, spread, and cumulative probabilities against the hand calculations from earlier chapters.

What you are looking at

One bar: the height of P(X = k)
Blue-highlighted bar: the count k you are focusing on
Mean / variance / standard deviation: how λ shapes the distribution
P(X ≤ k): cumulative probability of at most that many events

Try the presets in order — "Sparse (λ = 0.8) → Standard (λ = 2) → Busy (λ = 5) → Very busy (λ = 9)" — and you can track how the peak moves right and the shape smooths out.

Reference table for environments without the simulator

If JavaScript is disabled, or if you prefer to follow this chapter purely as text, the table below gives the key numbers for each preset. The simulator's "Mean", "Variance", "Mode candidate(s)", and "P(X = k)" readouts correspond to the columns here.

Preset	λ	Mean	Variance	Std. dev.	Relative spread (1/√λ)	Mode candidate(s)	e.g. `P(X = 2)`
Sparse	0.8	0.8	0.8	≈ 0.894	≈ 1.118	0	≈ 0.144
Standard	2	2	2	≈ 1.414	≈ 0.707	1 and 2	≈ 0.271
Busy	5	5	5	≈ 2.236	≈ 0.447	4 and 5	≈ 0.084
Very busy	9	9	9	3	≈ 0.333	8 and 9	≈ 0.0223

If you do use the simulator, working through the chapter while checking that the on-screen values match this table reinforces the connection between formula and figure.

Poisson simulator

As you move λ, the center, spread, and cumulative probabilities update on the spot.

Mean count λ 2.0 Highlighted count k 2 Max count shown 12

Mean2.0

Variance2.0

Std. dev.1.414

Relative spread (CV)0.707

Mode candidate(s)1 and 2

P(X = k)0.271

P(X ≤ k)0.677

P(X ≥ k)0.594

With λ = 2, the peak sits around 1–2 events, and 0 events still happens fairly often.

Four things worth trying

Press Sparse and confirm that the 0-event bar is the tallest.
Press Standard and read off the probability at k = 2.
Press Very busy and verify that both mean and variance equal 9.
Compare the relative spread between Standard and Very busy.

Hands-on 1 — Read values off the presets

Press the presets and confirm that the hand calculations match what the simulator shows.

Q1. With the Standard preset (λ = 2), what is P(X = 2)?

Q2. Under the Sparse preset (λ = 0.8), which count is the most likely?

0 events 1 event 2 events

Hands-on 2 — Shape of the Very busy preset

With the Very busy preset (λ = 9), read off the mean and variance.

Q1. What mean does the simulator display?

Q2. What variance does the simulator display?

Hands-on 3 — How relative spread changes with larger λ

Compare the relative spread std. dev. / mean between Standard (λ = 2) and Very busy (λ = 9).

Q1. As λ grows, how does the relative spread change?

Gets larger Gets smaller Stays the same

Key takeaways from Chapter 5

Moving λ shifts the peak and spread at the same time.
The mean, variance, and cumulative probabilities you computed by hand show up directly in the simulator.
Viewing the picture first and returning to the formula makes P(X = k) and P(X ≤ k) stick much more reliably.