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

Financial Ratio Analysis

Usefulness of Ratio Analysis

1. Measure of Profitability

Ratio analysis provides context for evaluating a company's profit, such as comparing it to previous periods or industry averages. Ratios like Gross Profit Margin, Net Profit Margin, and Expense Ratio help assess profitability and identify areas for improvement.

2. Evaluation of Operational Efficiency

Ratios like Turnover Ratios and Efficiency Ratios can reveal how effectively a company manages its assets and resources, helping to identify areas of inefficiency and potential cost savings.

3. Ensure Suitable Liquidity

Ratios such as the Current Ratio and Quick Ratio measure a company's short-term liquidity, indicating its ability to meet immediate cash obligations and... Continue reading "Financial Ratio Analysis: Uses, Benefits, and Limitations" »

Adding Like Terms in Algebra

When working with algebraic expressions, like terms are those that have the same pronumeral (letter) and the same powers. To add like terms, you combine their coefficients while keeping the pronumeral and its power unchanged.

For example:

5x + 3x

Because 'x' is the common pronumeral in this expression, you can simply add the coefficients (5 and 3) together. The sum is 8. Since 'x' is present in both terms, it must be included in the final answer. Therefore, the result is:

8x

Subtracting Like Terms in Algebra

Similar to addition, subtracting like terms involves combining terms that share the same pronumeral and powers. You subtract their coefficients while maintaining the pronumeral and its power.

For example:

5x - 3x

Since... Continue reading "Algebra Fundamentals: Terms, Patterns, and Equations" »

Vocabulary

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" »

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

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" »

Are X and Y independent random variables?

Solution: First we calculate the marginal pdfs

Note that fX,Y (x, y) 6= fX(x) × fY (y). So they are not independent.

What is the conditional distribution of X, given Y = .50? What is the probability that less than 50% of the initiative 1 surveys are returned, given that 50% of the initiative 2 surveys are returned?

(c) Calculate E(X/Y = y). What is the expected proportion of initiative 1 surveys returned, given that 50% of the initiative 2 surveys are returned?

Three prisoners are informed by their jailer that one of them has been 1/3 1/2

First we label the prisoners as P1, P2 and P3. Suppose P1 asks the jailer to tell him privately whether P2 or P3 will be set free. Now we define the following events.

Ai... Continue reading "Probability and Statistics Problems and Solutions" »

Descriptive and Inferential Statistics

Descriptive Statistics

Descriptive statistics involve organizing, summarizing, and presenting data. This can include using:

• Graphs
• Frequencies
• Tables

Inferential Statistics

Inferential statistics involve drawing conclusions from data.

Example: Fitness Center Data Analysis

A fitness center wants to know the average time clients exercise each week.

Key Concepts:

• Population: All clients in the fitness center
• Sample: A group of clients from the fitness center
• Parameter: Population mean amount of time clients exercise each week
• Statistic: Sample mean amount of time clients exercise each week
• Variable (X): The amount of time one client exercises in the center each week
• Data: Values for X (e.g., 4 hours, 6 hours, 10 hours)

Important

PRINCIPLES OF DESIGN

They affect content and message.

1. Proximity

Proximity basically means space; unfortunately, most people simply try to fill up empty space. Space should be organized, so, information can be easily understood.

2. Alignment

Alignment is like rulers or margins. We need to avoid to use more than one alignment, and to use the centered alignment.

3. Repetition

There are many ways to create repetition: -Bullets, bold fonts, color, line, a design element… Repetition unifies and strengthens. It creates visual interest.

4. Contrast

There are many ways to create contrasts: -Large/small type, warm/cold colors, old/new fonts, and horizontal/vertical… Contrast has 2 purposes: 1. Create interest on a page. 2. Aid organization on a page

5.