## Problem Description:

This Probability Theory homework provides essential functions and notations, an introduction to statistics, central tendency and standardization, correlation, probability concepts, random variables and expectation. It serves as a quick reference for key concepts in statistics and probability theory.

## Functions and Notations

**Function:**A function in computer science takes input (arguments) and produces output.**Summation Notation:**Denoted as ∑_(i=1)^n x_i, represents the sum of x1, x2, x3, and so on.**Commutative, Associative, and Distributive Properties:**Apply to sum and multiplication operations.

## Introduction to Statistics

**Statistics:**The study of collecting, analyzing, and interpreting sample data.**Descriptive Statistics:**Describes data features.**Inferential Statistics:**Draws conclusions about populations from sample data.**Schools of Thought:**Frequentist and Bayesian perspectives.**Psychometrics:**Focuses on psychological measurement.**Levels of Measurement:**Nominal, Ordinal, Interval, and Ratio.

## Central Tendency and Standardization

**Measure of Central Tendency:**Describes the "average" or "middle" of data. Includes mean, median, and mode.**Variability:**Measures how spread out data points are. Includes Variance, IQR, and Range.**Standardized Scores:**Transforms data into standard deviation units, allowing comparison between different normal distributions using z=(x-μ)/σ.

## Correlation

**Correlation Coefficient (Pearson r):**Measures the strength and direction of linear relationships between two variables. Ranges from -1 to 1.**Correlation is not Proof:**Correlation alone doesn't prove a relationship.**Linear Relationship:**Correlation measures linear relationships only, not curvilinear ones.**Correlation Formula:**r=Cov(X,Y)/√(Var(X)Var(Y)), where Cov(X,Y)=1/n ∑_i〖(X_i-X ̅)(Y_i-Y ̅)〗.**R Functions:**R provides predefined functions for basic calculations and distribution probabilities.

## Probability

**Sample Space:**The set of all possible outcomes in an experiment, denoted as S.**Random Event:**A subset of the sample space, denoted as E or F.**Mutually Exclusive Events:**Events E and F cannot occur together.**Probability Measure:**Maps random events to real numbers between 0 and 1.**Probability Axioms:**Three fundamental probability axioms.**Combinatorics:**The study of counting.**Permutations and Combinations:**Techniques for counting arrangements and sets of objects.**Conditional Probability:**P(E|F) measures the probability of E given that F has occurred.**Independence:**Two events E and F are independent if P(E∩F) = P(E) × P(F).

## Random Variable and Expectation

**Random Variable:**Maps random events to real numbers.**Discrete and Continuous Random Variables:**Different types based on possible value sets.**Probability Mass Function (PMF) and Probability Density Function (PDF):**Define the probability distribution.**Cumulative Distribution Function (CDF):**Computes the probability of a random variable being less than or equal to a specific value (quantile).**Parameters:**Characteristics that specify a random variable's distribution.**Expected Value (Mean):**Denoted by E(X), represents the long-run average value.**Variance:**Measures the squared distance of a random variable from its mean.**PMF for Bernoulli and Binomial:**Probability Mass Functions for these specific distributions.

This cheat sheet serves as a handy reference for fundamental concepts in statistics and probability, providing key notations, definitions, and formulas for various statistical tasks and calculations.