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

Question 1: Decimal Representation of a Fraction

Question: Consider the fraction 6/7. The decimal representation of this fraction is:

Answer: 6 ÷ 7 = 0.857142857... (repeating)

Question 2: Vaccinated to Unvaccinated Ratio

Question: If 60% of a population is vaccinated, what is the ratio of vaccinated to unvaccinated individuals?

Answer: 60% vaccinated → 60 : 40 → Simplified = 3 : 2

Question 3: Property Tax Calculation

Question: A property has been assessed at $225,000. The mill rate is 14.5. To find the property tax, you would multiply the assessed value by:

Answer: The mill rate of 14.5 means $14.50 per $1,000 of assessed value. To convert this to a decimal factor, divide by 1,000:

• 14.5 ÷ 1,000 = 0.0145
• Property tax = $225,000 × 0.0145 = $3,262.

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

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" »

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" »