Introduction to Statistics

First Class Notes: Randomness Observation: Noting occurrences. Realization: Understanding outcomes. Sample Space: The set of all possible outcomes; subsets are events with assigned probabilities. Sources of Variability: Reasons for randomness due to different possibilities. Events: Specific outcomes or sets of outcomes. Probabilities: Likelihoods assigned to events. Union of Sets: Combining all elements from sets A and B. Intersection of Sets: Common elements in sets A and B. Empty Set: A set with no elements. Sample Space (S): All possible outcomes. Useful Phrases: Disjoint: No common elements between sets. Exhaustive: All possible outcomes are covered. Second Class Notes Elements: Individual outcomes in a sample space. Events: Subsets of the sample space. Independent Events: P(A ∩ B) = P(A) · P(B). Conditional Probability: P(A|B) = P(A ∩ B)/P(B). Bayes' Theorem: Relates conditional probabilities. Third Class Notes Product Rule: For calculating joint probabilities. General Product Rule: An extension of the product rule. Summation Rule for Disjoint Sets: Summing probabilities of mutually exclusive events. General Formula for Non-Disjoint Sets: Summing probabilities considering overlap. Fourth Class Notes Core Concepts: Sample Space: All possible outcomes. Outcome: A single result. Event: A subset of outcomes. Realization: Actual outcome observed. Probability: Measure of likelihood. Axioms of Probability: Axiom One: Probability of any event is ≥ 0. Axiom Two: Probability of the sample space is 1. Axiom Three: For disjoint events, the probability of their union is the sum of their probabilities. Probability Rules: Union (Disjoint and General): Combining probabilities. Independence: No influence between events. Conditional Probability: Probability given another event. Intersection (Independent and General): Common probabilities. Fifth Class Notes Discrete Variables: Finite number of outcomes. Continuous Variables: Infinite outcomes within a range. Hybrid Variables: Combination of discrete and continuous characteristics. Sixth Class Notes Covariance (Cov(x, y)): Measure of how two variables change together. Covariance (Cov(aX, Y)): Influence of scaling one variable on their covariance.