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

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" »

Understanding Quartiles

Quartiles divide a set of ordered data into four groups with equal numbers of values. The three dividing points are Q1, the median (Q2), and Q3.

• Interquartile Range (IQR): Defined as Q3 – Q1, this represents the range of the middle half of the data. It provides a measure of spread by showing how closely the data are clustered around the median.
• Semi-interquartile Range: One half of the interquartile range.

Quartile Formulas

• Q2 (Median): (n+1) / 2
• Q1: (n+1) / 4
• Q3: 3(n+1) / 4

Outlier Formula

To identify outliers, use the following boundaries:

• Left Boundary: Q1 - 1.5(IQR)
• Right Boundary: Q3 + 1.5(IQR)

Deviation and Standard Deviation

Deviation tells you how far a single data value is from the mean (the difference between a value... Continue reading "Understanding Quartiles, Standard Deviation, and Percentiles" »

1. Type I and Type II Errors

Type I Error (α): Occurs when a researcher rejects a true null hypothesis (a "false positive").

Type II Error (β): Occurs when a researcher fails to reject a false null hypothesis (a "false negative").

The goal of statistical testing is to minimize both errors simultaneously.

2. Parametric vs. Non-Parametric Statistics

Parametric Tests: These assume data is normally distributed and use interval/ratio scales (e.g., t-test, ANOVA).

Non-Parametric Tests: These are "distribution-free" tests used for nominal/ordinal data or small samples (e.g., Chi-square, Mann-Whitney U).

Parametric tests are generally more powerful if their assumptions are met.

3. Null Hypothesis (H₀) vs. Alternative Hypothesis (H₁)

Null Hypothesis (H₀)

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,

Investment Portfolio and CAPM Practice Solutions

• Question 1: Portfolio beta (0.6)
• Question 2: Expected return under CAPM (7%)
• Question 3: Expected return with new allocation (6.5%)
• Question 4: Passive investment strategy (Zero alpha)
• Question 5: Net Asset Value calculation ($50)
• Question 6: Definition of NAV (Market value of assets minus liabilities divided by shares outstanding)
• Question 7: Portion of market risk (64%)
• Question 8: Mutual fund return after expenses (4.5%)
• Question 9: Net Asset Value City Street Fund ($40)
• Question 10: Impact on portfolio value (Decreases to $460 million)
• Question 11: Impact on shares outstanding (Decreases to 9 million)
• Question 12: New NAV after shares sold ($40)
• Question 13: Risk premium for stocks (60%)
• Question 14: Optimal

Common Statistical Biases

• Sampling bias: The sample was not representative of the population.
• Non-response bias: Only 24% returned surveys.

Sampling Techniques

• Simple Random Sampling (SRS): 1) Every member of the population has the same chance of being included (representative). 2) Members are chosen independently.
• Random Cluster Sampling: 1) Divide into smaller geographical sectors. 2) Take an SRS of sectors. 3) Count all samples in sectors and scale appropriately.
• Stratified Random Sampling: 1) Divide population into groups based on criteria like age or income. 2) Perform an SRS of each group and scale appropriately.

Data Variables and Distributions

• Variable Types: Categorical and Numeric (discrete and continuous).
• Relative frequency: Count / sample

Algebra and Trigonometry Problem Set

1. y = -6x³ + 4x² - x + 2
2. Vertical shrink by 1/2, horizontal translation 2 to the left, vertical translation 1 down
3. 4x⁴ - 11x³ + 7x² + 5x - 3
4. $713,476.20
5. Brian is designing a handrail for a staircase and needs to determine the length: 12
6. A meteorologist predicts a 42% chance of rain on Saturday and a 54% chance of rain on Sunday: 27%
7. a_n = 3n - 11
8. (-2, 6) lies on the terminal side of an angle θ in standard position: Sinθ = 3√10 is False, True, True
9. y = log(x - 2)
10. Find the sum of the series: 12
11. The table shows enrollment by gender as an undergraduate or graduate student: 515/2463 = 0.209

Zeros of Functions

12. Find the zeros of the function shown below:
f(x) = x² + 3x - 10: x = 2 and -5

Statistical Distributions

13. Normal... Continue reading "Algebra and Trigonometry Practice Problem Solutions" »

Set Difference Calculation

To find the set difference A - B, we identify all elements present in set A but not in set B.

Step-by-Step Subtraction

• Is 1 ∈ B? No. (Keep 1)
• Is 2 ∈ B? No. (Keep 2)
• Is 3 ∈ B? Yes. (Remove 3)
• Is 5 ∈ B? No. (Keep 5)
• Is 7 ∈ B? Yes. (Remove 7)
• Is 8 ∈ B? No. (Keep 8)

The remaining elements from set A are {1, 2, 5, 8}.

Symbolic Logic

In symbolic logic, the word "but" functions like "and," indicating that both conditions occur simultaneously. To write "He is rich but not generous" in symbolic form:

• p: "He is rich"
• q: "He is generous"
• ¬q: "He is not generous"
• ∧: The conjunction operator

Logic Symbol Reference

Logarithm Calculations

To find the value of log 360,... Continue reading "Step-by-Step Solutions for Mathematical Problems" »