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

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

This step-by-step analysis covers Problem Sets 6-9, emphasizing key concepts from Problem Sets 7 and 8, essential for your final exam.

Problem Set 6: Product Differentiation & Merger Impacts

1. Why Bertrand Does Not Equal Marginal Cost in Reality

• Firms may experience:

  • Capacity constraints

  • Brand loyalty (differentiated products)

  • Reputational concerns or switching costs

2. Bertrand Competition with Differentiated Products

• Demand:

  • Q_M = 1000 - 200P_M + 100P_B

  • Q_B = 1000 - 200P_B + 100P_M

• Steps:

  1. Plug in rival's price to derive inverse demand.

  2. Derive Marginal Revenue (MR); set MR = Marginal Cost (MC) = 4.

  3. Solve for the best response price.

  4. Set both best responses equal to solve for the Nash Equilibrium (NE).

  5. Calculate quantity, profit, and price-cost margin.

Chapter 2: Predetermined Overhead Rate

Predetermined Overhead Rate = Estimated Total Manufacturing Overhead (MOH) / Estimated Total MOH Driver (e.g., Direct Labor hours, Direct Labor costs, Machine Hours)

Prime Cost = Direct Materials + Direct Manufacturing Labor

Conversion Cost = Direct Manufacturing Labor + Indirect Manufacturing Overhead

Cost Accumulation: Data is collected in an organized way (also known as cost pools).

Cost Assignment: Systematically links an actual cost pool to a distinct cost object (e.g., Tires, engine, labor assigned to car cost).

Activity Base: Examples include kilometers driven in a car, units produced, units sold, machine hours.

Product Cost: Costs tied to creating a product (Direct Materials, Direct Labor, Manufacturing... Continue reading "Cost Accounting Essentials: Key Concepts and Calculations" »

Statistical Measures and Causal Inference Concepts

Measures of Dispersion and Relationship

Variance

Variance: Estimates how far a set of numbers (random) are spread out from their mean value.

Covariance

Covariance: The relationship between two variables.

• Cov = 0: Unsure of the relationship.
• Cov > 0: Suggests Y will be above average when X is above average.
• Cov < 0: Suggests Y will be below average when X is above average.

The formula for variance is often expressed as: $\mathbb{E}[X^2] - (\mathbb{E}[X])^2$ (where $\mathbb{E}$ is the Expected Value).

The formula for covariance between two variables $X$ and $Y$ is: $\mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$

Pearson's Correlation Coefficient

Standardizes covariance between -1 and 1:

Pearson’s

AdaBoost: Adaptive Boosting Algorithm Explained

AdaBoost (Adaptive Boosting) is a classic and widely used boosting algorithm that focuses on correcting the errors of preceding weak learners (typically decision trees). It works by iteratively adjusting the weights of the training data points.

How AdaBoost Works

1. Initial Weights: AdaBoost starts by assigning equal weights to all the training data points.
2. Train a Weak Learner: A "weak" learner (a model that performs slightly better than random chance, like a decision stump) is trained on the dataset using the current weights.
3. Calculate Error and Performance: The error rate of the weak learner is calculated based on the instances it misclassified. A measure of the weak learner's performance (often called