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

Key Concepts in Engineering Economics

Engineering Economics is the science dealing with quantitative analysis techniques for selecting the most preferable alternative from several technically viable options.

Fundamental Principles

Four fundamental principles must be applied in all engineering economic decisions:

• The time value of money
• Differential (or incremental) cost and revenue
• Marginal cost and revenue
• The trade-off between risk and reward

Core Terminology Explained

Ethics

A set of principles that guides a decision-maker in distinguishing between right and wrong.

Market Interest Rate

The interest rate quoted by financial institutions, which refers to the cost of money for borrowers or the earnings from money for lenders.

Interest Rate

The cost, or price,

Managerial Decision-Making and Business Analytics

Types of Managerial Decisions

To effectively plan, coordinate, and lead, managers make several types of decisions:

• Strategic Decisions: Address high-level issues and the overall direction of the organization. They define future goals and are long-term and complex.
• Tactical Decisions: Focus on how to achieve the goals and objectives set by the strategy. These are typically made by mid-level management for the medium term.
• Operational Decisions: Pertain to day-to-day operations. They are made by operations managers and are often simple and routine.

The Decision-Making Process (DMP)

A structured approach to decision-making involves several key steps:

1. Identify and define the problem.
2. Determine the criteria

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)

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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:

Bar Graph: Annual Sales of Product A and B

The bar chart illustrates the annual dollar sales of Product A and Product B for the years 2015, 2016, and 2017. As can be seen in the graph, between 2015 and 2017 sales of Product A were higher than sales of Product B. In 2015, sales of Product B were slightly lower than Product A, and in 2016 sales of Product A reached 80,000 USD while sales of Product B only reached 50,000 USD. For 2017, both Product A and Product B had a slight growth, increasing their sales by 10,000 USD compared to the previous year. Overall, we can see that sales of both products have grown in the last three years; however, the product that generates the most revenue is Product A.

Line Chart: Six-Year Sales Trend

The graph shows... Continue reading "Annual Sales Trends and Household Water Usage Data" »

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