Distributions (normal, binomial, Poisson)

Why distributions matter

Distributions model uncertainty and randomness.

• Normal: measurement noise, averages
• Binomial: number of successes in N trials (conversions)
• Poisson: counts over time/space (arrivals, events)

Picking the right distribution

  flowchart TD
  A["What are you modeling?"] --> B["Continuous measurement
(height, latency, noise)"]
  A --> C["Count of successes
in a fixed number of trials"]
  A --> D["Count of events
over time or space"]
  B --> E["Normal"]
  C --> F["Binomial"]
  D --> G["Poisson"]

Normal distribution

• Parameters: mean (\mu), std (\sigma)
• Symmetric bell curve

import numpy as np
 
x = np.random.normal(loc=0, scale=1, size=10_000)
print(x.mean(), x.std())

Binomial distribution

• Parameters: trials (n), success prob (p)
• Example: 100 visitors, conversion probability 0.03

import numpy as np
 
samples = np.random.binomial(n=100, p=0.03, size=10_000)
print(samples.mean())

Poisson distribution

• Parameter: rate (\lambda)
• Example: number of support tickets per hour

import numpy as np
 
samples = np.random.poisson(lam=5, size=10_000)
print(samples.mean())

Visualize it

The three distributions have distinct shapes: the normal is a smooth continuous bell, the binomial counts successes in a fixed number of trials, and the Poisson counts rare events over an interval. Seeing them side by side makes the differences click:

Practical tip

When unsure:

• Plot a histogram of your data.
• Start with simple candidates (normal vs count distributions).
• Check mean/variance: for Poisson, mean ≈ variance.

Check your understanding

🧪 Try It Yourself

Exercise 1 – Simulate a normal distribution

Press Run to see what your code prints.

Exercise 2 – Binomial trials

Press Run to see what your code prints.

Exercise 3 – Poisson arrivals

Press Run to see what your code prints.

Next

Continue to Sampling and the Central Limit Theorem to see why sample means behave so predictably no matter which distribution you started from.