Contents

Calculus Based Statistics

2021-08-26

Contents

Based on Probability and Statistics for Engineering and the Sciences 9e.

Descriptive Statistics

Description	Equations
Sample mean	$\bar{x} = \dfrac{1}{n}\sum\limits_{i=1}^n x_i$
Sample median	$\tilde{x} = \footnotesize \begin{cases}\text{middle ordered value} &\text{if odd} \cr \text{average of two middle ordered values} &\text{if even} \end{cases}$
Deviation	$x_i - \bar{x}$
Sum of squared deviation	$\begin{aligned} S_{xx} &= \textstyle \sum(x_i - \bar{x})^2 \cr &= \textstyle \sum x_i^2 - \dfrac{1}{n}(\sum x_i)^2 \cr &= \textstyle \sum x_i^2 - n(\bar{x})^2 \end{aligned}$
Sample variance	$s^2 = \dfrac{S_{xx}}{n-1}$
Sample standard deviation	$s = \sqrt{s^2}$
Properties of sample variance	$x_i' = x_i + c \implies s_{x'}^2 = s_{x}^2 \newline x_i' = cx_i \implies s_{x'}^2 = c^2s_{x}^2$

Probability

Basic Probability

Notations

Description	Equations
Sample space	$\mathcal{S}$
Event	$A$
Union (or)	$A \cup B$
Intersection (and)	$A \cap B$
Complement (not)	$A'$
Null event	$\varnothing$
Disjoint (mutually exclusive)	$A \cap B = \varnothing$
Probability of event $A$	$P(A)$

Axioms

Description	Equations
Non-negativity	$P(A) \ge 0$
Probability of event in sample space	$P(\mathcal{S}) = 1$
Addition of infinite disjoint events	$\bigcap\limits_{i=1}^\infty A_i = \varnothing \implies P\left(\bigcup\limits_{i=1}^\infty A_i\right) = \sum\limits_{i=1}^\infty P(A_i)$

Properties

Description	Equations
Null event and zero probability	$P(\varnothing) = 0$
Probability of event and its complement	$P(A) + P(A') = 1$
Maximum probability	$P(A) \le 1$
Addition of disjoint events	$P(A \cup B) = P(A) + P(B)$
Addition of any two events	$P(A \cup B \cup C) = P(A) + P(B) - P(A \cap B)$
Addition of any three events	$P(A \cup B) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$

Conditional probability

Description	Equations
Conditional probability of $A$ given $B$ occured	$P(A\vert B) = \dfrac{P(A \cap B)}{P(B)}$
Probability of event intersection	$P(A \cap B) = P(A \vert B)P(B)$
Law of total probability	$\bigcap\limits_{i=1}^{k} A_i = \varnothing \implies P(B) = \sum\limits_{i=1}^{k} P(B \vert A_i)P(A_i)$
Bayes' Theorem	$P(A \vert B)={\dfrac {P(B \vert A)P(A)}{P(B)}}$
Bayes' Theorem	$P(A_j \vert B) = \dfrac{P(A_j \cap B)}{P(B)} = \dfrac{P(B \vert A_j)P(A_j)}{\sum\limits_{i=1}^{k} P(B \vert A_i)P(A_i)}$

Independence

Description	Equations
Independent events	$P(A \vert B) = P(A)$
Dependent events	$P(A \vert B) \not= P(A)$
Probability of independent event intersection	$P(A \vert B) = P(A) \iff P(A \cap B) = P(A)P(B)$
Mutually independent events	$P(\bigcap\limits_{i=1}^k A_i) = \prod\limits_{i=1}^k P(A_i)$

Counting

Description	Equations
$k$ -tuples	Ordered collection of $k$ elements
Product rule for $k$ -tuples	A set of $k$ -tuples with probability $p_i$ for $i$ th element has $\prod p_i$ possible $k$ -tuples.
Permutations (ordered subset) of size k formed from n individuals	$P_{k, n} = \dfrac{n!}{(n-k)!}$
Combinations (unordered subset) of size k formed from n individuals	$\displaystyle C_{k, n} = \binom{n}{k} = \dfrac{P_{k, n}}{k!} = \dfrac{n!}{k!(n-k)!}$

Discrete Random Variables and Probability Distributions

Random variables

Description	Equations

Inferential Statistics