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
Inferential Statistics