Stop 02 · Math & Statistics for AITopics · 8/13

Probability Distributions

In short

Heights, exam marks, clicks per ad, orders per minute: each follows a typical shape. Knowing the common distributions helps you spot weird data and pick the right model or loss.

Normal, Bernoulli, binomial and Poisson distributions describe how data and model outputs are shaped — essential for understanding model outputs and statistical tests.

The purple blob pours chana through a wooden pegboard, forming a bell-shaped pile in the slots below.

Think of it like this

Chana through a pegboard

Pour chana from the top of a board full of pegs. Each one bounces left or right at random. Most land in the middle slots, fewer at the edges, and a bell-shaped pile forms. That pile's shape is a distribution: it shows how likely each outcome is.

Watch it happen

The common distributions

Distributions

Discrete

Continuous

Bernoulli

Binomial

Poisson

Normal

Uniform

1/5Discrete outcomes are countable, like clicks. Continuous ones can be any value in a range, like height.

The big ideas

What you need to know

A distribution is a shape of chance

It tells you which values are common and which are rare. Mean and spread describe it in two numbers.

The 68-95 rule

For a normal distribution, about 68% of values fall within 1 standard deviation of the mean, and about 95% within 2.

Counts are not bell curves

Counts can't be negative and are often skewed. Poisson or binomial usually fits them better than normal.

Models output distributions

A classifier's softmax gives a probability for each class. Those numbers form a distribution that adds to 1.

See it in code

Walk through the code

python

1import numpy as np2rng = np.random.default_rng(0)3 4heights = rng.normal(loc=165, scale=7, size=10_000)    # cm5print(heights.mean().round(1), heights.std().round(1))  # about 165.0 7.06print(np.mean(np.abs(heights - 165) < 14))              # about 0.957 8views = rng.binomial(n=1, p=0.03, size=10)     # Bernoulli: clicked or not9clicks = rng.binomial(n=100, p=0.03, size=5)   # clicks per 100 views10orders = rng.poisson(lam=4, size=5)            # orders per minute11print(views, clicks, orders)

1/310,000 heights from a normal with mean 165 and standard deviation 7.

default_rng is numpy's recommended random generator. Sampling from a distribution and checking its numbers is a quick way to build intuition.

Save yourself the pain

Common mistakes

Oops

Assuming every column is normally distributed.

Do this instead

Plot a histogram first. Incomes, waiting times and counts are often skewed.

Oops

Using a normal model for counts and getting negative predictions.

Do this instead

Use a distribution that fits counts, like Poisson, so predictions stay at zero or above.

Your mission

Draw the pegboard

~20 min

Quick check

You want to model how many customers walk into a shop each minute. Which distribution fits best?

Go deeper

1Learn the shape and use case of the normal, Bernoulli, binomial and Poisson distributions.
2Plot each distribution in Python with different parameters to build intuition.
3Connect distributions to real ML use cases (e.g. logistic regression outputs a Bernoulli probability).