Notes, summaries, assignments, exams, and problems for Mathematics

Chapter 1: Understanding Variables

Types of Variables

• Categorical: Smoker (current, former, no)
• Ordinal: Non, light, moderate, heavy smoker (ordered categories)
• Quantitative: BMI, Age, Weight (numerical measurements)

Key Definitions

• Observation: Measurements are made (individual or aggregate).
• Variable: The generic characteristic we measure (e.g., age).
• Value: A realized measurement (e.g., 27).

Chapter 2: Statistical Studies

Surveys: Census and Sampling

• Goal: Describe population characteristics.
• Census: Attempts to reach the entire population (costly, time-consuming).
• Sampling: Uses a sample of the population (allows for inferences, saves time and money).
• Simple Random Sampling: Based on probability.
• Issues with Sampling: Under-coverage, volunteer bias,

Parametric Inference Fundamentals

The probability distribution of the population under study is known, except for a finite number of parameters. Its goal is to estimate those parameters. Examples include the T-test and ANOVA.

Non-Parametric Inference Basics

The distribution of the population is not known. It is used to test the assumptions of parametric methods, for example, to check if the population distribution is normal.

What is a Statistic?

A random variable function of the sample that does not depend on the unknown parameter.

Understanding Estimators

A statistic whose values are acceptable for estimating an unknown parameter.

Unbiasedness in Estimation

We do not allow systematic overestimation or underestimation of the parameter, which would result... Continue reading "Statistical Inference & Hypothesis Testing Concepts" »

De Morgan's Law

De Morgan's Law: (Flip if the union is true)

, image of set: [min, max]; one-to-one: horizontal line test; Onto: Image must equal domain; Bijective: one-to-one and Onto

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Possible Outcomes and Probability Calculations

• Repetition formula: n^k
  • Example: 5 awards (k) and 30 students (n), with no limit to awards per student.
• Permutation formula: P(n, k) = n! / (n - k)!
  • Example: Each student gets 1 award, so the number of students decreases by one each award.
• No overlap probability: P(n, k) / repetition formula
• Arrangements: a = slots → a! can be multiplied by arrangements within slots
• Die sum probability:
  • List combinations that lead to the sum for each die.
  • If a die is rolled multiple times, each combination has (rolls)! permutations.
  • Add

Probability Distributions for Discrete Events

The following table matches common scenarios to their appropriate probability distributions:

AdaBoost: Adaptive Boosting Explained

AdaBoost is one of the simplest and earliest boosting algorithms. The main idea behind AdaBoost is to combine many weak learners (models that do slightly better than random guessing) into one strong learner.

It works by training multiple models one after another. After each model, the algorithm checks which data points were predicted wrong. It then gives more importance (weight) to those wrongly predicted samples so that the next model focuses more on correcting those mistakes.

Each new model tries to fix the errors made by the previous ones. At the end, all models are combined using weighted voting to make the final prediction. This helps improve accuracy and reduces errors.

Key Characteristics of AdaBoost

• Combines

Fundamental Auction Concepts

Payoff: A bidder's payoff is their valuation for the item minus the price paid.

Social Surplus: This is the sum of the surpluses of all participants. The formula is: Seller's Surplus (p) + Winner's Surplus (v - p) + Loser's Surplus (0). Here, v is the winner's valuation and p is the price paid. Social surplus is maximized, and the auction is considered efficient, if the winner is the bidder with the highest valuation.

Types of Auctions

English Auction

This is a type of ascending auction where an auctioneer announces prices, and bidders accept or reject them.

• Winner: The last remaining bidder.
• Price: The second-highest price or bid.
• Information Revealed: The auctioneer learns the valuations of all bidders except for the

Floating Point Systems & Numerical Error

A Floating Point (FP) System represents numbers as: x = ± (d₀ + d₁/β + d₂/β² + ... + d_t-1/β^(t-1)). The Unit Roundoff (u) is defined as ε_machine/2, where fl(1 + ε) > 1.

Rounding to Nearest

When rounding to the nearest representable number, fl(x) = x(1 + ε) where |ε|.

IEEE 754 Standard for Floating Point

Normalized Numbers

If the exponent (e) is not equal to 0, it's a normalized FP number. The value is x = (-1)^sign ⋅ β^{(e - offset)} ⋅ (1.d₁ d₂...d_t-1).

Denormalized Numbers

If the exponent (e) is 0, the number is denormalized. The value is x = (-1)^sign ⋅ β^{(e - offset + 1)} ⋅ (0.d₁ d₂...d_t-1). The sticky bit 0 is free because it is always determined by the value of exponent e.

Exceptional Values

• If

Map Generalization

Types of Symbols

• Line Symbols: Represent real-life objects with a linear path.
• Point Symbols: Represent objects occurring at a single point on Earth's surface using a dot.
• Area (Polygon) Symbols: Represent real-life objects spread over Earth's surface using geometric shapes.

Generalization Techniques

Reality contains too much information for a single 2D map. Generalized geometry and content make a map useful. A good map suppresses less important information to highlight what needs to be seen.

• Selection: Only relevant line, point, and area features are chosen.
• Classification: Grouping similar features and using a common symbol to represent them.
• Simplification: Reduction of unnecessary detail.
• Smoothing: Smoothing out abruptly joined

Boolean Algebra Fundamentals

Formulating Expressions: SOP and POS

Sum of Products (SOP) Formulation Steps

1. Circle rows in the truth table where the output $Y = 1$.
2. Identify the minterms corresponding to the circled rows.
3. Sum (OR) the selected minterms to form the final expression.

Product of Sums (POS) Formulation Steps

1. Circle rows in the truth table where the output $Y = 0$.
2. Identify the maxterms corresponding to the circled rows.
3. Multiply (AND) the selected maxterms to form the final expression.

Order of Evaluation in Boolean Algebra

Operations are evaluated in the following sequence:

1. Parentheses
2. NOT (Complementation)
3. AND (Multiplication)
4. OR (Addition)

Fundamental Boolean Laws and Theorems

• Identity Laws

  • $A + 0 = A$
  • $A \cdot 1 = A$

• Null Laws

  • $A + 1 = 1$
  • $A \cdot