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())

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())

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())

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.

🧪 Try It Yourself

Exercise 1 – Simulate a normal distribution

Exercise 2 – Binomial trials

Exercise 3 – Poisson arrivals

Next

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