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

Fundamental Theorems in Probability

The Law of Large Numbers

The Law of Large Numbers is a theorem in probability that describes the average behavior of a succession of random variables as the total number of variables increases. The theorem provides sufficient hypotheses to affirm that the sample average converges to the average of the expectations of the random variables involved.

Chebyshev's Theorem

Chebyshev's Theorem provides an upper bound on the probability that values fall outside a certain range from the average.

Bernoulli's Theorem

Bernoulli's Theorem is a special case of the Law of Large Numbers, which specifies that the approximate frequency of an occurrence converges to the probability p of that occurrence as the experiment is repeated.... Continue reading "Core Principles of Probability and Statistical Inference" »

Case Study 1: Sales Team Head Settlement

Employee Details & Salary Structure

Settlement Calculation Steps

Unpaid Salary

• Period: November 1 - November 3 (3 days)

Monthly Salary = 1,090 + 320 + 90 = 1,500 Euros/month

Unpaid Salary for 3 days = (1,500 / 30) * 3 = 150 Euros

Pro-rata

Tangible Fixed Assets Definition

Tangible fixed assets are a set of rights and resources intended to remain in the business for more than one financial period.

Types of Fixed Assets

• Tangible-Intangible Assets: These assets have a physical entity but are classified as intangible assets or rights subject to economic evaluation.
• Tangible Assets: These assets have a physical entity and are classified as real or personal property, except those that must be classified in other subgroups.

Understanding Depreciation

Depreciation refers to the systematic allocation of the cost of tangible items over their useful life due to three primary reasons:

1. The use made of these elements, which causes deterioration.
2. The passage of time, whether the asset is used or not.

VAT Fees Eligible for Deduction

The following fees can be deducted:

• Submission-entry of goods from another taxable person
• Imports of goods
• Intra-Community acquisitions
• Fees relating to cases of reverse charge

Partial Deduction or Pro Rata

This applies when a company performs two types of activities: operations with deduction and operations without deduction.

The consequence is that only a portion or percentage of the taxes charged can be deducted.

What percentage? To determine the coefficient, calculate the following:

(Sum of output-delivery deduction in the year) / (Sum of all delivery-end of the year)

To calculate the deduction, multiply the percentage by the taxes charged.

Dynamics During the Exercise: Provisionally apply the percentage from the previous... Continue reading "VAT Deduction Rules: Fees, Partial Deduction, and Investment" »

Frequency Distribution

1) Loose Data: Data collected that has not been organized numerically. For example, heights of males and females obtained from a survey, arranged alphabetically.

2) Ordination: An arrangement of numerical data in ascending or descending order. The difference between the largest and smallest number is called the data range.

Example:

Data Management Data Range
2.1 1 A: 10 to 1
7.6 2 R: 9
4.9 4
8.10 6 R: range of the data.
7
8
9
10

3) Frequency Distribution: When dealing with large amounts of loose data, it's useful to distribute them into classes or categories and determine the number of individuals belonging to each category, often called a class.

A tabular distribution of data by type or category with its corresponding... Continue reading "Understanding Frequency Distributions: Data Organization" »

Types of Research

Descriptive and Explanatory Research

Descriptive Research: Aims to show a situation or phenomenon.

Explanatory Research: Seeks to uncover the truth of a hypothesis.

Transverse and Longitudinal Research

Transverse Research: Involves studying different variables at a specific point in time.

Longitudinal Research: Involves studying variables over an extended period.

Retrospective and Prospective Research

Retrospective Research: Observation units are classified based on the presence or absence of an effect, and the frequency of causal factors is recorded.

Prospective Research: Individuals are selected based on the presence or absence of exposures (inputs), and the frequency of the effect is recorded.

Case-Control Studies

• Combines advantages

Descriptive Statistics

Classification and description of a dataset.

Statistical Inference

Interpreting results using descriptive techniques for decision-making.

Statistic

A numerical summary or descriptive feature of a total sample of research.

Frequency Polygon

A line graph usually plotted on a frequency histogram.

Experiment

A controlled procedure carried out to determine, demonstrate, or test a hypothesis; a process that yields an outcome.

Median

The midpoint of a data set.

Probability

Measures or quantitatively determines the possibility that an event occurs.

Objective Probability

Probability obtained through numerical calculations or empirical observations.

Classic Approach

Assumes that all events are equally likely to occur.

Event

A possible outcome or... Continue reading "Essential Statistics and Probability Terms Defined" »

Review Guidelines for Software Quality

1. We review the product, not the producer.
2. Ensure an agenda for review items is prepared and maintained.
3. Limit challenges and discussions to stay focused.
4. State problems when detected, but do not attempt to solve them during meetings.
5. Reviewers should include meeting participation in their work plan.
6. Utilize these meetings for training less experienced team members.

Formal Approaches to Software Quality Assurance

In recent decades, there has been a growing tendency to adopt a more formal approach to software quality assurance. This perspective views a program not merely as a mathematical object. By defining a rigorous mathematical syntax and semantics for all languages, and an equally rigorous approach to requirements... Continue reading "Enhancing Software Quality: Methodologies and Practices" »

The Book of Compensation: A Payroll Record

"The Book of Compensation" is a crucial record where all amounts related to employee compensation payments are documented. This includes Social Security contributions, legal deductions, and agreed taxes.

Payroll Components: Understanding Salary Calculations

Employee salaries are comprised of three main calculation types:

• Taxable Income
• Non-Taxable Income
• Discounts/Deductions

Taxable Income Components

Taxable income includes elements such as basic salary, overtime, bonuses, gratuities, and commissions. These are considered taxable because their total amount is subject to estimated discount rates and pension laws.

Base Salary

The base salary is the fixed monetary amount paid for regular periods. By law, these... Continue reading "Employee Compensation Structure: Payroll and Benefits Explained" »

Business Documents

Commercial documents, also called "proof," constitute the evidence of any business operation conducted by a company. Thus, each proof tests a type of transaction. It is known that any operation involves at least two parts. For each of them, it will have a different meaning (as discussed later: buyer or seller, debtor or creditor, or collector payer, etc.).