Probability I

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Definitions, key results and distributions, unit by unit.

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Unit 1: Probability spaces, axioms, and rigorous counting
Unit 2: Conditional probability and independence
Unit 3: Random variables and distribution functions
Unit 4: Expectation, moments, and transforms
Unit 5: Joint distributions and conditional expectation
Unit 6: Modes of convergence, laws of large numbers, and CLT

Unit 1

Probability spaces, axioms, and rigorous counting

Probability spaces and axioms

Probability space

A triple $(\Omega, \mathcal{F}, P)$ : a sample space $\Omega$ of outcomes, a $\sigma$ -algebra $\mathcal{F}$ of events (contains $\Omega$ , closed under complements and countable unions), and a probability measure $P$ on $\mathcal{F}$ .

Axioms of probability

(1) $P(A) \ge 0$ for every event $A$ . (2) $P(\Omega) = 1$ . (3) Countable additivity: if $A_1, A_2, \dots$ are pairwise disjoint, then $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$ .

Complement rule

P(A^c) = 1 - P(A)

In particular $P(\emptyset) = 0$ .

Monotonicity

A \subseteq B \implies P(A) \le P(B)

Inclusion–exclusion (two events)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Inclusion–exclusion (n events)

P\Big(\bigcup_{i=1}^n A_i\Big) = \sum_{k=1}^{n} (-1)^{k+1} \sum_{1 \le i_1 < \cdots < i_k \le n} P(A_{i_1} \cap \cdots \cap A_{i_k})

Union bound (Boole's inequality)

P\Big(\bigcup_{i} A_i\Big) \le \sum_{i} P(A_i)

Holds for finite or countable collections, disjoint or not.

Continuity of probability

A_1 \subseteq A_2 \subseteq \cdots \implies P\Big(\bigcup_{n} A_n\Big) = \lim_{n \to \infty} P(A_n)

Likewise, if $A_1 \supseteq A_2 \supseteq \cdots$ then $P\left(\bigcap_n A_n\right) = \lim_n P(A_n)$ .

Counting

Equally likely outcomes

P(A) = \frac{|A|}{|\Omega|}

Only when $\Omega$ is finite and every outcome has the same probability.

Multiplication principle

If a procedure has $r$ stages and stage $j$ can be done in $n_j$ ways regardless of earlier choices, the whole procedure can be done in $n_1 n_2 \cdots n_r$ ways.

Ordered, with replacement

n^k

Sequences of length $k$ from $n$ symbols.

Ordered, without replacement (permutations)

n(n-1)\cdots(n-k+1) = \frac{n!}{(n-k)!}

With $k = n$ this is $n!$ , the number of orderings of $n$ objects.

Unordered, without replacement (combinations)

\binom{n}{k} = \frac{n!}{k!\,(n-k)!}

Unordered, with replacement (stars and bars)

\binom{n+k-1}{k}

Multisets of size $k$ from $n$ types; equivalently, solutions of $x_1 + \cdots + x_n = k$ in nonnegative integers.

Multinomial coefficient

\binom{n}{n_1, n_2, \dots, n_r} = \frac{n!}{n_1!\,n_2!\cdots n_r!}, \quad n_1 + \cdots + n_r = n

Ways to split $n$ labelled objects into groups of sizes $n_1,\dots,n_r$ .

Binomial theorem

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}

Vandermonde's identity

\binom{m+n}{k} = \sum_{j=0}^{k} \binom{m}{j}\binom{n}{k-j}

Unit 2

Conditional probability and independence

Unit 3

Random variables and distribution functions

Unit 4

Expectation, moments, and transforms

Unit 5

Joint distributions and conditional expectation

Unit 6

Modes of convergence, laws of large numbers, and CLT

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