Probablity & Statistics

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Last updated: 22 Sept 2025

Basics

• Experiment : Repeatable task with well defined outcomes

• Sample Space : All possible outcomes of an experiment

• Event : A subset of sample space of experiment $E\subseteq S$

Set Theory

1. $\ A \subseteq B => A\ subset\ of\ B $
2. $\ A = B => A\subseteq B \ and \ B\subseteq A$
3. $\ Empty\ set (\phi) => Set\ with\ no\ element => \phi\ is\ contained in every set$

Operations with sets

$A\cup B = { x: x \in A\ or\ x \in B }$
$A\cap B = { x: x \in A\ and \ x \in B }$
$A^c = { x : x\notin A }$

Suppose we have an $\Gamma$ be indexing set and we have {$A_\alpha ,\ \alpha \in \Gamma$} be collection of sets indexed by $\Gamma$

then , $\bigcup\limits_{\alpha\in\Gamma} A_\alpha = { x : x \in A_\alpha\ for\ some\ \alpha \in \Gamma}$

and $\bigcap\limits_{\alpha\in\Gamma} A_\alpha = { x : x \in A_\alpha\ for\ every\ \alpha \in \Gamma}$

1. A and B are disjoint if $A \cap B = \phi$, A and B are Mutually Exclusive

2. $A_1, A_2, ..... $ are pairwise disjoint if $A_i \cap A_j = \phi \qquad \forall\ i \neq j$

3. $A_1, A_2, ..... $ is a partition of S if :

4. $A_1, A_2,...$ are pairwise disjoint

5. $\bigcup\limits_{i} A_i = S$

Sigma Algebra

• Collection of subsets of S

• Satisfying

• $\phi \in \mathcal{B}$

• if $A \in \mathcal{B},\ then\ A^c \in \mathcal{B}$

• if $A_1, A_2, A_3,... \in \mathcal{B}$, then $\bigcup\limits_{i=1}\limits^{\infty} A_i \in \mathcal{B}$, (closed under countable unions).

• ${\phi, S}$ is trivial sigma algebra associated with S.

Probablity Function

• Probablity function for pair (S, $\mathcal{B}$) is defined for P:$\mathcal{B}\ \rightarrow \mathbb{R} $ if it satisfies below Axioms of Probablity:

• $P(A) \geq 0 \qquad \forall A\in \mathcal{B}$

• P(S) = 1

• if $A_1, A_2, A_3,... \in \mathcal{B}$, are pairwise disjoint (countable) then $P(\bigcup\limits_{i=1}\limits^{\infty}) = \sum\limits_{i=1}\limits^{\infty}P(A_i) $)

• Properties of Probablity fn :

• P($\phi$) = 0

• P(A) $\le$ 1

• P($A^c$) = 1 - P(A)

• 0 $\le P(A) \le$ 1

• $P(B\cap A^c) = P(B) - P(A\cap B)$

• $P(A \cup B) = P(A) + P(B) - P(A\cup B)$

• if $A \subseteq B$ => P(A) $\le$ P(B)

Conditional Probablity & Independence

• Conditional P : Decision under influence of info

• P(A|B) = Probablity of A given B has occured

• $P(A|B) = \frac{P(A\ \cap\ B)}{P(B)}$

• Multiplication Rule : $P(A\cap B) = P(A|B)P(B) = P(B|A)P(A)$

• Independent Events : $A, B \subseteq S\ are\ independent\ iff$

• $P(A|B) = P(A) \quad <=> \quad P(B|A) = P(B)$

Bayes Theorum

• use prior probablities to calculate posterior probablities

• $P(A_j | B) = \frac{P(B|A_j)*P(A_j)}{\sum\limits_{i-1}\limits^{n} P(B|A_i) * P(A_i)} $, all probablities on rhs are prior probs

Counting

• Fundamental Theory of Counting

• Let T be task of perfoming k tasks sequentially : $T_1, T_2,....T_k$ that can be permormed in $n_1, n_2,....n_k$ ways,

• Total no of ways to perform T = $n_1n_2...*n_k$