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

Econometric Notation: Residuals and Estimates

Residuals are denoted by e (Sample Regression Function, SRF) and the true error term by u (Population Regression Function, PRF).

• Coefficients: Estimated coefficients (e.g., $\hat{\beta}_0$ and $\hat{\beta}_1$) in the SRF are denoted with hats (caps). True coefficients ($\beta_0, \beta_1$) in the PRF are without hats.
• Dependent Variable: The predicted value $\hat{Y}_t$ (the value derived from the regression equation without the residual) is denoted with a hat. The actual observed value $Y_t$ (which includes the residual or error term) is without a hat.

Fundamentals of Hypothesis Testing

Hypothesis testing makes inferences about the validity of specific economic (or other) theories from a sample of the... Continue reading "Econometric Hypothesis Testing and Regression Model Specification" »

MATLAB Cable Tension and Cost Optimization

This script calculates cable tensions and optimizes costs for a structural system by iterating through coordinate positions.

Initialization

ACMatrix = zeros(51,51);
ABMatrix = zeros(51,51);
ADMatrix = zeros(51,51);
CostMatrix = zeros(51,51);
countery = 0;
MinCost = 1;

A = [6 2 1]';
B = [3.5 0 -0.5]';
D = [0 -1 5.5]';
F = -[160 20 -30]';

rab = B - A;
rad = D - A;
ead = rad/norm(rad);
eab = rab/norm(rab);

Iterative Calculation

The following loop calculates tensions and costs across a range of Y and Z coordinates:

• C: Point position of the system.
• Tensions: Solved using the unit vector matrix.
• Cost: Calculated based on cable length and tension.

for y = 1:.1:6
    countery = countery + 1;
    counterz = 0;

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Cost Tool Information

The Cost Tool provides the following information:

• Manufacturer
• Overall Cost (Total)
• Profit Percentage
• Overall Weight
• SKU Number
• Shipping Location (Town and State)

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Accounting Fundamentals: Key Concepts & Transactions

Essential Accounting Terminology

Core Concepts

• Accounting: Practice, activity, action.
• Accountancy: Theory, technique.
• Accountant: Person, profession.
• Account: Record, ledger, report, put the $ on.

Key Roles and Statements

Roles

• CFO: Chief Financial Officer.
• Accountant
• Bookkeeper

Financial Statements

• Balance Sheet: Financial statement showing the financial situation of the company.
• Profit & Loss Statement: Financial statement to show the profit earned or losses incurred in a determinate period.

Debits, Credits, and Balances

• DEBIT: When you move the $ out.
• CREDIT: Transfer the $ to another account.
• To Debit Account: To take $ out of account.

Assets and Liabilities

Assets

• Assets: Any property owned by a person or firm.
• Intangible Asset: Asset that you cannot

What is Statistics?

Statistics is a set of tools designed to analyze data and deduce information about a population from a given sample.

The Three-Step Process of Statistics

1. Sampling and Design of the Experiment: Take a sample (or many) from the population, make observations about the sample, and turn them into numerical data.
2. Descriptive Statistics: Analyze the data to get information about the sample.
3. Statistical Inference: From the data, deduce information about the whole population.

Context is Crucial

The context of a statistical study is crucial in interpreting the results. A population is a set of individuals (people, cases, etc.) that we want to analyze. A sample is a subset of the population. A variable is an aspect or characteristic of the... Continue reading "Statistics: A Guide to Data Analysis and Population Inference" »

Frequency Tables for Data Organization

A frequency table is the easiest way to organize data. To construct a frequency table from the data, essentially we have to count how many times each event occurs among the data. Depending on the type of variable, we can add more or less information to the frequency table.

• Absolute frequency: Count how many times each value appears in the table.
• Relative frequency: Count how many times each value appears in the table, divided by the size of the sample.
• Cumulative absolute frequency: The sum of all previous absolute frequencies (e.g., how many people own 3 cars or less?).
• Cumulative relative frequency: The sum of all previous relative frequencies (e.g., what proportion of people own 3 cars or less?).

Discrete

Visualizing Discrete and Continuous Data

We can use bar charts to describe data from a discrete variable. In this case, we can perform additional analysis: first, create a frequency table, then draw the charts.

Cumulative Bar Charts

In addition to the standard bar chart, we can draw a cumulative bar chart:

• The standard bar chart allows for comparison among groups.
• The cumulative bar chart displays percentages.
• Note: Cumulative bar charts require an inherent order among outcomes, so they are not suitable for categorical variables.

Histograms for Continuous Variables

Histograms are the equivalent of bar charts for continuous variables:

• To draw a histogram, you must first group the data, similar to building a frequency table.
• Key Difference: In a histogram,

MONEY

Budgeting and Expenses

• Budget: A plan for how to spend money.
• Grant: Money given by the government for a particular purpose, often education.
• Loan: Money borrowed from a bank or other lender, usually with interest.
• Fee: Money paid for a professional service (e.g., lawyer, consultant).
• Fare: Money paid to travel by bus, train, taxi, etc.

Savings and Investments

• Savings: Money set aside for future use.
• Donation: Money given to a charity or other organization.
• Deposit: A portion of a larger payment made upfront.
• Will: A legal document that specifies how a person's assets will be distributed after their death.
• Lump sum: A single payment of a large amount of money.

Financial Terms

• Fine: Money paid as a penalty for breaking a rule or law.
• Installment: A

Decision Making and Problem Solving

Defining the Process

Problem Solving: Identifying the difference between the actual and desired state of affairs, and taking action to resolve the difference.

Decision Making: Defining the problem, identifying alternatives, determining criteria, evaluating alternatives, and choosing an alternative.

Types of Decision Problems

Single-Criterion Decision Problem: Finding the best solution based on one criterion.

Multicriteria Decision Problem: Finding the best solution considering multiple criteria.

Key Concepts in Decision Making

Decision: The selected alternative.

Model: A representation of a real object or situation.

• Iconic Model: A physical replica of a real object.
• Analog Model: A physical representation, but doesn'