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

Caesar Cipher: Formal Representation

Plain alphabet: P = {sequence of plaintext letters}. Key: k ∈ {i | 0 ≤ i ≤ 25}. If k = 25, the shift maps a → z, b → a, and so on. Encryption: E(p) = (p + k) mod 26. Decryption: D(c) = (26 + c − k) mod 26.

Attacking the Caesar Cipher

Common methods to solve or attack a Caesar (shift) cipher include:

1. Brute force: Try all possible keys (0–25) and inspect the results.
2. Statistical (frequency) analysis: Use letter frequency distributions of the language to infer likely mappings.

Frequency Analysis: Basic Idea

Certain letters appear more frequently than others in a given language. By comparing ciphertext letter frequencies to natural language frequencies, you can match ciphertext characters to likely plaintext

Hypothesis Testing

Statistical Test Selection

1. If the population standard deviation is unknown and the sample size is less than 30: t-test

2. If the population standard deviation is known and the sample size is less than 30: t-test

7. Hypothesis test on population mean; n = 25; σ = 2.5: z-test

8. Hypothesis test on population mean; n = 50; s = 7.2: z-test

18. Test statistic for sample size above 30: z-test

19. Test statistic when population standard deviation is known: z-test

20. Test statistic when population standard deviation is unknown: t-test

21. When to use the t-test: I and II

24. Optimal sample size for z-test: Equal to or larger than 30

Hypotheses and Significance

3. H₀: μ = 30

4. H₁: μ > 30

5. No

9. False: The alternative hypothesis typically... Continue reading "Hypothesis Testing: A Concise Statistical Method Reference" »

What is Data Science?

• An interdisciplinary field combining statistics, computer science, and business knowledge.
• Its goal is to extract valuable insights and knowledge from data (both structured and unstructured).
• It answers key business questions: what happened, why, what will happen, and what to do about it.
• The process involves collecting, cleaning, processing, analyzing, and communicating data insights.

Statistical Inference: Making Educated Guesses

• It's the process of using sample data to make educated guesses or draw conclusions about a much larger population.
• Essentially, it lets you make generalizations about a whole group based on a smaller part of it.

Key Goals of Statistical Inference

• Estimation: To guess the value of a population parameter

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.

Quick Graph Identification

• Holes: Represented by open circles.
• Vertical Asymptotes: Represented by dashed vertical lines.
• Horizontal or Slant Asymptotes: Represented by dashed lines.
• Note: The graph approaches but never touches the asymptotes.

Understanding Domain

• Exclude values that make the denominator equal to zero.
• Even if a factor cancels, the value is still excluded from the domain.

Example: (x + 3) / (x(x + 3))
Domain: x ≠ -3, x ≠ 0

Final One-Pass Checklist

1. Factor and cancel.
2. Find holes.
3. Find vertical asymptotes.
4. Find x-intercepts.
5. Find the y-intercept.
6. Find horizontal or slant asymptotes.

Exponential and Logarithmic Functions

Exponential: f(x) = a · b^(x - h) + k (where b > 0, b ≠ 1)
Logarithmic: f(x) = a · log_b(x - h) + k
Note: Logarithms... Continue reading "Mastering Rational, Exponential, and Logarithmic Functions" »

Executive Summary of Regression Models

• Simple Linear Regression: On average, for every 1-unit increase in [X], the expected [Y] changes by β1 units (95% CI: …).
• Multiplicative Model: On average, a 1-unit increase in [X] multiplies the median [Y] by exp(β1), resulting in a 100·(exp(β1)–1)% change (95% CI: …).
• Power Law/Elasticity: A 1% increase in [X] is associated with a β1% change in [Y] (95% CI: …).
• Categorical Variable: Students in Group A scored on average β1 units higher or lower than those in Group B (95% CI: …).
• Categorical Variable (3-Group): After adjusting for [X], students taught with Method 2 scored on average β1 units higher than those with Method 1; Method 3 scored β3 units lower.
• Interaction: For Group A, a 1-unit

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

Understanding Bookkeeping

Bookkeeping is the systematic recording of financial transactions of a business in books of accounts on a day-to-day basis.

Objectives of Bookkeeping

• Systematic Record: To keep a complete and permanent record of all business transactions.
• Ascertain Profit or Loss: Helps in finding profit or loss at the end of the accounting period.
• Ascertain Financial Position: Helps in knowing assets and liabilities of the business.
• Legal Evidence: Acts as proof in legal matters.

Advantages of Bookkeeping

• All transactions are properly recorded.
• Management can take better decisions.
• Provides information about profit, loss, assets, and liabilities.
• Makes auditing easier.
• Helps compare past and present performance.

Accounting Fundamentals

Accounting... Continue reading "Bookkeeping and Accounting Fundamentals Explained" »

Handshaking Lemma

In any undirected graph, the sum of the degrees of all vertices is twice the number of edges.

Formula: Σdeg(v) = 2|E|

Euler's Formula for Planar Graphs

For planar graphs, the relationship between vertices (V), edges (E), and regions (R) is defined as:

V - E + R = 2

Sum of Degrees and Odd Vertices

The sum of the degrees of all vertices in any graph is always even because each edge contributes 2 to the total sum. Furthermore, the number of vertices with an odd degree must always be even.

Graphs with No Odd Degree Vertices

If all vertices in a graph have an even degree, the graph is Eulerian, meaning it contains an Eulerian circuit.

Complete Graphs

A complete graph with n vertices, denoted as K_n, has an edge between every pair of distinct... Continue reading "Essential Graph Theory Formulas and Concepts" »