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 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.
- 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.