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

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

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

Data Science, Machine Learning, and Artificial Intelligence

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

1. Understand the Problem

The first step to solving a two-step algebraic equation is to clearly write down the problem. This helps you visualize the solution process. For our example, we will work with the equation: -4x + 7 = 15.

2. Isolate the Variable Term Using Addition or Subtraction

The next step is to isolate the variable term (e.g., "-4x") on one side of the equation and the constants (whole numbers) on the other. To achieve this, you'll use the Additive Inverse. Find the opposite of the constant term on the same side as the variable. In our example, the constant is +7, so its additive inverse is -7.

Subtract 7 from both sides of the equation to cancel out the "+7" on the variable's side. Write "-7" below the 7 on the left side and below... Continue reading "Mastering Two-Step Algebraic Equations" »