# 🎲 Cheatsheet: Mathematics for Machine Learning 3: Probability and Statistics for Data-Driven Models

By Better Wise

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A lookup card for the whole course. Every formula is tied to a concrete dataset question.

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## Module 1: Probability Foundations for Data

### 1.1 Axioms, Conditional Probability and Independence

**The three Kolmogorov axioms** (event space $\Omega$, probability measure $P$):

| # | Axiom | Statement |
| --- | --- | --- |
| 1 | Non-negativity | $P(A) \ge 0$ for every event $A$ |
| 2 | Normalization | $P(\Omega) = 1$ |
| 3 | Additivity | If $A, B$ disjoint, $P(A \cup B) = P(A) + P(B)$ |

**Consequences you use constantly:**
- $P(A^c) = 1 - P(A)$
- $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- $P(\emptyset) = 0$

**Conditional probability:**
$$
P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0
$$

**Multiplication rule:** $P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)$

**Independence:** $A \perp B$ iff $P(A \cap B) = P(A)P(B)$, equivalently $P(A \mid B) = P(A)$.

> **Warning:** 
> Disjoint (mutually exclusive) is NOT independent. If $P(A)>0$ and $P(B)>0$ and they are disjoint, then $P(A \mid B) = 0 \ne P(A)$, so they are dependent.

**Law of Total Probability** (partition $B_1, \dots, B_k$ of $\Omega$):
$$
P(A) = \sum_{i=1}^{k} P(A \mid B_i)\,P(B_i)
$$

**Dataset question:** *In `sessions.csv`, what fraction of sessions convert given the user arrived from a paid ad?* → condition on the `channel` column.

### 1.2 Expectation and Variance on a Real Dataset

**Random variable:** a function from outcomes to numbers (e.g. `order_value` per session).

| Quantity | Discrete | Continuous |
| --- | --- | --- |
| Expectation | $E[X] = \sum_x x\,P(X{=}x)$ | $E[X] = \int x\,f(x)\,dx$ |
| Variance | $\mathrm{Var}(X) = E[(X - \mu)^2]$ | same definition |

**Shortcut form:** $\mathrm{Var}(X) = E[X^2] - (E[X])^2$

**Linearity of expectation** (always holds, no independence needed):
$$
E[aX + bY + c] = aE[X] + bE[Y] + c
$$