Probability Distributions: Normal, Binomial & Poisson

Interpret distribution models Explained before reaching for a formula

Daily defect counts on a production line may be roughly Poisson-like if events are independent and the rate is stable, while the number of customers who convert out of 1,000 visitors is naturally modeled as a binomial count.

• A probability distribution assigns probabilities to possible outcomes and provides a model for uncertainty.
• The binomial distribution models a fixed number of independent success/failure trials, while Poisson often models event counts in an interval.
• The normal distribution is continuous and symmetric, but real data should not be assumed normal without checking.
• Expected value and variance summarize a distribution, yet tail behavior can still matter for risk-sensitive decisions.

Calculate a small case and verify it in Python

For distribution models, predict one visible result before each snippet runs—a row count, shape, statistic, ranking or error pattern—then compare it with the printed checkpoint.

# distribution models · example 1
from scipy.stats import binom, poisson, norm
print('P(7 successes of 10):', binom.pmf(7, 10, 0.6))
print('P(3 arrivals):', poisson.pmf(3, 2.5))
print('95th percentile:', norm.ppf(0.95))

distribution models checkpoint 1: Run the cell and inspect the printed values before changing anything.

# distribution models · example 2
import pandas as pd
import numpy as np

df = pd.read_csv('practice.csv')
x = df.select_dtypes('number').iloc[:,0].dropna()
print('n=', len(x), 'mean=', round(x.mean(),2), 'median=', round(x.median(),2))

distribution models checkpoint 2: n, mean and median are printed from the first numeric column.

# distribution models · example 3
q1, q3 = x.quantile([0.25,0.75])
iqr = q3 - q1
print('std=', round(x.std(ddof=1),2), 'IQR=', round(iqr,2))

distribution models checkpoint 3: Standard deviation and IQR provide two different views of spread.

# distribution models · example 4
rng = np.random.default_rng(42)
boot = [rng.choice(x, size=len(x), replace=True).mean() for _ in range(1000)]
print(np.percentile(boot,[2.5,50,97.5]).round(2))

distribution models checkpoint 4: The three values are the bootstrap 2.5th, 50th and 97.5th percentiles.

# distribution models · example 5
x2 = pd.concat([x, pd.Series([x.max()*5])], ignore_index=True)
print('mean change:', round(x2.mean()-x.mean(),2))
print('median change:', round(x2.median()-x.median(),2))

distribution models checkpoint 5: The mean usually moves more than the median when one extreme value is added.

Stress-test the result against assumptions and outliers

Self-check with answers

Practise interpretation, not only arithmetic