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

Survival Analysis Fundamentals

Understanding Survival Analysis

The main objective of survival analysis techniques is to determine differences between two or more treatments applied to a set of individuals. Each individual receives a particular treatment, and the effect (response) is measured by the occurrence of a specific event of interest (e.g., default) and the time elapsed from the start of observation until the aforementioned event occurs. Survival analysis techniques apply to data with the following characteristics:

• The dependent variable (or response variable) is the time that elapses until the individual experiences a specific event of interest, often termed death. Therefore, while the individual does not experience the event of interest,

Statistical Characteristics and Variables

A statistical character is a property that enables the classification of individuals in a population. These can be:

• Quantitative: When they are measurable.
• Qualitative: If they are not measurable.

Statistical variables represent the different values a quantitative variable can take. They are classified as:

• Discrete: When they take specific, isolated values.
• Continuous: When they can take any value within a range.

Data Frequency Analysis

Frequency data provides essential information:

• Absolute frequency: The number of times a value repeats.
• Relative frequency: The ratio between the frequency of a value and the total number of data points.
• Cumulative absolute frequency: The sum of the absolute frequency of a value

Contribution Bases in Special Situations

Temporary Incapacity, Pregnancy Risk, and Childbirth

The obligation to contribute remains for situations of temporary incapacity, risk during pregnancy, and enjoyed periods of maternity leave, although they involve a cause of suspension of employment. The contribution base for the various contingencies is determined as follows:

a) Basis for Common Contingencies

The applicable contribution base for common contingencies will be determined by the month preceding the date of disability, the onset of risk during pregnancy, or entitlement to maternity leave, according to the following rules:

• In the event of wages being paid on a daily basis or if, having that character, the worker has remained registered with the

Understanding Bills of Exchange

A bill of exchange is a crucial financial instrument in commercial transactions. This document outlines its key characteristics, the parties involved, and related financial operations.

Key Characteristics of a Bill of Exchange

• All rights and obligations arising from a bill of exchange must be explicitly stated within the document itself.

Essential Data for a Bill of Exchange

A bill of exchange must include the following critical data:

1. Denomination of the bill of exchange: Clearly stating it is a bill of exchange.
2. Order to pay a certain sum of money: An unconditional order to pay a specific amount.
3. Name of the drawee: The person or entity ordered to pay.
4. Maturity date: The date when payment is due (not always essential)

Statistical Attributes and Variables

Attributes are qualitative characters. The different categories of an attribute are called modalities. These conditions should be exhaustive and mutually exclusive.

Variables are quantitative traits. Variables can also be expressed as attributes. The measurement of a variable results in a value, which can be discrete or continuous.

Defining Population and Samples

• Population: A collective of individuals or elements that have certain common characteristics and are the object of observation and statistical study.
• Subpopulation: A subset of elements of the population that has a certain characteristic which differentiates them from others.
• Sample: A subset of elements of the population that aims to represent the entire

Random Variables and Probability Concepts

Random variable.
Meaning: So far we have studied general characteristics of probability spaces (Ω, F, P), where Ω is the sample space, F is a σ-algebra and P is a probability function. Often we are not interested in studying the events of the σ-algebra themselves, but one or several numerical characteristics associated with the outcome of the experiment. This may be, for example, the event of selecting a student from the university, the probability that your grade is greater than or equal to 5 (likely to pass), or when establishing a new tax the probability that an individual's annual income does not exceed 20,000 euros. These numerical characteristics, which under certain conditions we can associate... Continue reading "Random Variables: Discrete, Continuous & Probability Theory" »

Valuation Criteria

Historical Cost

For Assets: The purchase price or production cost. This includes the amount of cash paid or payable, plus the fair value of any other consideration given for the acquisition. All costs directly related to the acquisition and necessary to bring the asset to operating condition are included.

For Liabilities: The value corresponding to the consideration received in exchange for incurring the debt. In some cases, it is the amount of cash expected to be paid to settle the liability in the ordinary course of business.

Fair Value

The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between knowledgeable, willing market participants at the measurement date. It is determined... Continue reading "Understanding Asset and Liability Valuation Concepts" »

Understanding Science and the Scientific Method

Science is knowledge obtained through observation and reasoning, systematically structured. The scientific method consists of observation, which leads to a hypothesis that must be proven or disproven.

Observation and Measurement

Observation is the activity of a living being that detects and registers data using the senses as the primary tools. The term can also refer to any data collected during this activity.

Measurement is the comparison of a magnitude with another, arbitrarily taken as a reference standard to express how many times it is contained. A standard of measurement is a pattern of actions isolated to serve as a foundation for creating a unit of measurement. While many units have patterns,... Continue reading "Fundamentals of Scientific Measurement and Vector Analysis" »

Técnicas de Recolección de Datos

La recolección de datos es el proceso mediante el cual el investigador se vincula con el objeto de estudio para extraer información. Esta puede provenir de fuentes primarias o secundarias:

• Primarias: Técnicas como la observación (hojas de registro) o la entrevista (formularios).
• Secundarias: Técnicas de fichaje (registros bibliográficos, lectura, citas) e instrumentos como fichas bibliográficas o resúmenes.

La Bibliografía

Es la base esencial para la planificación y el desarrollo de la investigación. Debe incluirse al final del trabajo o en el cuerpo del texto para facilitar la búsqueda de información. Todo texto consultado debe registrarse como bibliografía utilizada y ordenarse alfabéticamente... Continue reading "Metodología de Investigación: Técnicas y Citas" »

Cauchy's Mean Value Theorem

Suppose f and g are two functions such that:

• f and g are continuous on the closed interval [a, b].
• f and g are differentiable on the open interval (a, b).
• For all x in the open interval (a, b), g'(x) ≠ 0.

Then there exists a number z in the open interval (a, b) such that:

[f(b) - f(a)] / [g(b) - g(a)] = f'(z) / g'(z)

Proof of the Theorem

Proving that g(b) is not equal to g(a)

First, we show that g(b) ≠ g(a). We use a proof by contradiction; assume that g(b) = g(a). Since g satisfies the conditions of the Mean Value Theorem, there is some number c in (a, b) such that:

g'(c) = [g(b) - g(a)] / (b - a)

If we assume g(b) = g(a), then g(b) - g(a) = 0, which implies g'(c) = 0. This contradicts the third condition of our hypothesis,... Continue reading "Cauchy's Mean Value Theorem and Its Proof" »