# Assessment on Essential Statistics and Probability Concepts

This comprehensive cheat sheet is your ultimate guide to mastering the fundamental concepts of statistics and probability. Whether you're a student, researcher, or data enthusiast, this resource provides a clear and concise reference for essential functions, notations, and key principles in the world of data analysis. Explore topics ranging from central tendency to correlation, probability, random variables, and expectation. With this cheat sheet, you'll have a quick and accessible reference at your fingertips for all your statistical and probability needs.

## 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.