4. Statistics and Probability

Modeling is a process of creating a mathematical representation of certain real-world scenarios. This falls under the statistics field, which requires some terminology in prior.

Terminology	Definition
Population	Entire collection of individuals about which we want to draw conclusions
Census	Information collected from the whole population
Sample	Subset of population which could be chosen at random to avoid bias
Survey	Collection of information from a sample
Data	Information about individuals in a population
Categorical variable	Describes a particular quality or characteristics which can be divided into different categories
Numerical variable	Describes a characteristic which has a numerical value that can be counted or measured. There are two types: discrete and continuous.
Discrete variable	Takes exact values and is often a result of counting
Continuous variable	Takes numerical values within a certain range, and is often a result of measuring
Random variable	A variable that is subject to random variations so that it can take on multiple different values, each with an associated probability. Often takes $X$ , $Y$ etc.
Parameter	Numerical quantity that measures some aspect of the population
Statistics	Quantity calculated from data gathered from a sample
Mean	The average of a data set: $µ=\frac{\sum f}{n}$
Median	The middle value in a sorted list of numbers. Given $n$ items, $\tilde{x} = \frac {n+1}2$ th item
Mode	Most frequently occurring data
Interquartile Range IQR	The difference between the upper quartile (75 percentile) and the lower quartile (25 percentile).
Variance $\sigma ^2$	The squared deviation from the mean of a random variable. $\sigma ^2= \frac {\sum _{i=1}^{n}(x_i-µ)^2}{n}$
Standard Deviation $\sigma$	Statistical measurement that analyzes the distance of the data from the mean. This is a square root of the variance. $\sigma =\sqrt{\frac{ \sum _{i=1}^{n}(x_i-µ)^2}{n}}$

4.1 Statistics

There are two types of probability, experimental and theoretical.

Terminology	Definition
Number of trials	Total number of trials the experiment is repeated.
Outcomes	Different results possible for one trial of the experiment.
Frequency	Number of times of a certain outcome being observed.
Relative Frequency	Frequency represented as a fraction against the total number of trials.
Cumulative Frequency	The total of a frequency and all frequencies in a frequency distribution until a certain defined class interval.
Sample space $U$	Set of all possible outcomes of an experiment.

4.2 Probability

Discrete Distributions

If

X

is a random variable with sample space {

x_1, x_2, ..., x_n

} and corresponding probabilities

p_i

, then {

p_1, p_2, ..., p_n

} describes the probability distribution of

X

. We call

P(x) = P(X=x)

as the probability mass function (PMF).

4.3 Distributions

Random Variable	Definition
Discrete random variables	A random variable assigns numbers to the possible outcomes of a random experiment. A discrete random variable, $X$ , has a finite or countably infinite set of distinct values. Discrete probability distributions are representable in a table due to its categorical nature. The probabilities must add up to 1. The probability distribution of a discrete random variable describes the probabilities associated with each value. For a random variable $X$ with possible values { $x_1,x_2,...,x_n$ } and corresponding probabilities { $p_1,p_2,...,p_n$ }: • $0\leq p_i\leq 1$ • $\sum_{i=1}^np_i=1$ Example: If $P(X)=\frac x{2+1}$ for $x=1,2,3,4$ , then $P(X)$ is a valid probability mass function.
Expectation (Mean)	The expected value $E(X)$ of a discrete random variable $X$ with values $x_i$ and probabilities $p_i$ : $E(X)=\sum _{i=1}^nx_ip_i$ $E(X)$ is the mean of the probability distribution of $X$ , denoted $\mu$ .
Variance	$Var(X)$ measures the spread of the random variable around its mean: $Var(X)=\sum_{i=1}^np_i(x_i-\mu)^2$ Alternatively: $Var(X)=E(X^2)-(E(X))^2$ Standard deviation $\sigma=\sqrt{Var(X)}$
Linear transformations	• $E(aX+b)= aE(X) + b$ • $E(X + Y) = E(X) + E(Y)$ • $Var(aX+b) = a^2Var(X)$ • $Var(X + Y) = Var(X) + Var(Y)$ if $X$ and $Y$ are independent. • $\sigma(X) = \|a\|\sigma(X)$

Random Variable	Definition
Continuous random variables	A random variable $X$ is continuous if its cumulative distribution function is continuous. If we have: $F_X(x)=\int_{-∞}^xf_X(t) dt$ $f_X$ is the probability density function (PDF) of $X$ . Note that PDF and PMF are fundamentally different objects, as PDFs can be greater than 1. Below, for simplicity, we denote PDF as $f(x)$ . A continuous random variable $X$ can take any value within an interval $[a,b]$ . PDF $f(x)$ describes the probability distribution: • $P(c\leq X\leq d)=\int_c^df(x)dx$ • $f(x)\geq 0$ for all $x$ • $\int_a^bf(x)dx=1$
Mode, median, mean	Mode: The value $x$ at which $f(x)$ is maximum. Median: The value $m$ such that $\int_a^mf(x)dx=\frac 12$ . Mean (Expected Value): $\mu =E(X)=\int_a^bxf(x)dx$
Variance and standard deviation	Variance: $\sigma^2=Var(X)=\int_a^b(x-\mu)^2f(x)dx=\int_a^bx^2f(x)dx-\mu^2$ Standard deviation: $\sigma=\sqrt{Var(X)}$

4.4 Random variables

done

https://forms.gle/QynSnLyHR8HQ66557

4.4 Random Variables