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

Coordinate Geometry Formulas

For points A(x₁, y₁) and B(x₂, y₂), the distance formula is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Functions and Calculus Basics

• Difference quotient: f(x+h) - f(x) / h
• X-intercepts: Not imaginary, written as (x, y).
• Solutions/Roots/Zeros: Can be imaginary, written as x = ___.

Financial and Parent Functions

• Compound interest: A = P(1 + r/n)^nt (r must be a decimal).
• Continuous compound interest: A = Pe^rt
• Parent function y = x²: Domain: all real numbers, Range: y ≥ 0.

• Parent function y = √x: Domain: inside ≥ 0, Range: y ≥ 0.

Transformations and Analysis

Key: For transformations, ensure x inside parentheses is always positive; factor negatives out front.

• h(x) = -3^2(x+4): Left 4, flip

Axiom

Statement that is considered to be self evident and assumed to be true without any proof or demonstration.

Theorem

Statement that has been proved on the basis of previously established statements, such as other theorems and generally accepted statements such axioms.

Corollary

Statement that follows with little or no proof required from an already proven statement.

Lemma

Mathematical result that is useful in establishing the truth values of some other results.

Trivial proof

The statement on the proof is trivial if we can prove that Q(x) is true for all x in S, then for all x in S, p(x) => q(X) is true regardless of the truth value of p(x).

Vacuous proof

The statement on the proof is vacuous if we can prove that P(x) is true for all x in S, then... Continue reading "Mathematical Proofs and Definitions" »

Activity 10: Conditional Probability

Objective

To explain the computation of conditional probability of a given event, assuming event B has already occurred, using the example of throwing a pair of dice.

Method of Construction

• Paste a white paper on a piece of plywood of a convenient size.
• Create a square and divide it into 36 unit squares (1cm each).
• Write the pairs of numbers representing the outcomes.

Demonstration

1. The figure displays all possible outcomes of the experiment, representing the sample space.
2. Suppose we find the conditional probability of event A given event B has occurred, where A is "a number 4 appears on both dice" and B is "4 appears at least once." We must find P(A|B).
3. From the figure: Number of outcomes favorable to B = 11; Number

Selecting Samples

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

A random sample of size n from an infinite population is a sample selected such that the following conditions are satisfied.

Point Estimation

By making the preceding computations, we perform the statistical procedure called point estimation. We refer to the sample mean x as the point estimator of the population mean m, the sample standard deviation s as the point estimator of the population standard deviation s, and the sample proportion p as the point estimator of the population proportion p. The numerical value obtained for x, s, or p is called the point... Continue reading "Understanding Sampling, Estimation, and Hypothesis Testing in Statistics" »

Types of Data

Population and Sample

When dealing with large datasets, it's often impractical to analyze every single data point. In such instances, we collect data from a subset of the population known as a sample.

Quantitative and Categorical Data

Quantitative data represent numerical values and allow for arithmetic operations like addition, subtraction, multiplication, and division. Examples include height, weight, and temperature.

Categorical data represent categories or groups and cannot be manipulated with arithmetic operations. Examples include gender, hair color, and country of origin. We can summarize categorical data by counting the number of observations or computing the proportions of observations in each category.

Cross-Sectional and

Understanding Imaginary Numbers

√-25 has no real solution because (-5)² = 25 and (5)² = 25, never -25.

Definition of the Imaginary Unit

√-1 = i (imaginary unit). Therefore, √-25 = √25 * √-1 = 5i.

In general: √-any = √any * i. Example: √-200 = i√200 = 10i√2.

Powers of i

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

This pattern repeats every four powers. Example: i¹⁵ = i¹² * i³ = -i.

Complex Numbers

Complex numbers are a combination of real and imaginary numbers. Standard form: a + bi (Real part + Imaginary part).

Operations with Complex Numbers

• Absolute Value: |4+3i| = √4²+3² = √16+9 = √25 = 5
• Conjugate: The conjugate of (4+3i) is (4-3i).
• Addition: (4+3i) + (2-i) = 6+2i
• Subtraction: (4+3i) - (2-i) = 2+4i
• Multiplication: (4+3i)*(2-i)

Fundamental Statistical Concepts

Estimates and Population Values

Estimates are statistics collected from samples and used to describe population values.

Scales of Measurement

Understanding the different scales of measurement is crucial for data analysis:

• Nominal Scale of Measurement: This is the simplest form of measurement, used when the dependent variable is qualitative. It categorizes data based on the name of some physical or psychological quality or characteristic rather than a numerical score.
• Ordinal Scale of Measurement: This is the simplest quantitative scale used to measure the dependent variable. Data are ranked in some order (such as highest to lowest, biggest to smallest, most important to least important), but it does not measure the

Floating-Point Arithmetic and Stability

Eigendecomposition: Symmetric & Positive Definite (PD)

Exponent bias: b = 011111111111 = 1023

Grampian matrix:

Maximum exponent: maxp = 1023

Minimum exponent: minp = -1023 - 51

L2 Induced Matrix Norm:

Precision range: 2**-52 <= s < 2

Note on stability: A stable algorithm cannot overcome an ill-conditioned system; it can only guarantee precision when the system is not ill-conditioned.

Special Floating-Point Values:

• np.Inf: 0111111111110
• np.NaN: 0111111111111
• np.Inf - np.Inf: NaN
• np.Inf / np.Inf: NaN
• np.Inf * 0: NaN
• np.NaN + any: NaN

Backward Error: Back-calculating from the computed result to find the equivalent perturbation of the input.

Analytical inverse and the Woodbury matrix identity are both effective... Continue reading "Numerical Linear Algebra: Matrix Computations and Stability" »

Clear, Consistent, Long-Term Goals

Google's mission statement is to organize the world's information and make it universally accessible and useful.

Inditex

One clear aspect of Inditex's strategy is its founder's vision for the future and unwavering belief in his work. A key goal was to create the best logistics system in the market—an innovative formula that placed garments in stores in under fifteen days, regardless of location.

Profound Understanding of the Competitive Environment

Tesla

Elon Musk's innovative technology company, Tesla, is known for producing high-quality, cutting-edge vehicles with high-end and creative features. Tesla has been changing the auto industry with its full self-driving capability cars: Model S, Model X, and Model... Continue reading "Long-Term Vision & Competitive Analysis: Case Studies" »