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

Fundamentals of Statistical Analysis

Population and Sampling

All statistical study refers to a set of elements called the population. When the population is very large, and the goal is to focus on only a part of it, a sample is taken. This sample can represent the entire population or be a partial survey.

Analyzing Representative Samples

A representative sample can be analyzed using two main methods based on strata (subgroups) of the population:

• Equal Allocation: The sample is taken such that the number of elements selected from each stratum of the population is the same.
• Proportional Allocation: The representative sample is taken in proportion to the number of elements in each stratum.

Characteristics and Variables

The characteristics (or qualities)... Continue reading "Statistical Concepts: Sampling, Variables, and Data Visualization" »

Financial Mathematics

Core Definitions

• Annuity: A series of equal payments made at regular intervals to pay off a debt or build a fund.
• Capital: The principal amount of money agreed upon for an investment or loan.
• Interest: The amount earned or paid for the use of money, directly proportional to capital, time, and the interest rate.
• Compound Interest: Interest calculated on the initial principal and also on the accumulated interest of previous periods; its growth is exponential.
• Simple Interest: Interest calculated only on the initial principal; the capital remains constant each period, resulting in arithmetic growth.
• Amount: The total sum of money to be paid or received at the end of an agreed period, calculated as capital plus interest.
• Deadline:

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,

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" »

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" »

La gestion des stocks adéquate est celle qui minimise les niveaux de stocks.

Une rupture de stock survient lorsque le niveau de stocks d'un article particulier est insuffisant pour les besoins de l'entreprise, l'empêchant de respecter ses engagements envers les clients ou l'obligeant à interrompre le processus de production. Pour éviter cette situation, il convient de disposer d'un stock de sécurité.

Le niveau du stock de sécurité est le niveau le plus bas de stocks que l'entreprise devrait avoir pour que la production ne soit pas interrompue.

Coûts de la gestion des stocks

La détention de stocks entraîne divers coûts :

• Coût d'approvisionnement : C'est le coût d'achat du produit. Cp = pQ
• Coût de passation de commande : C'est le coût

Sampling Stage

Sample Size:

• Depends on population size
• The type of sample
• Feature (parameter) of the population that is analyzed
• The maximum permissible error in the estimation of the parameters

The Form: The information collected can be classified as:

• Acts or behaviors that can be compared
• Information: analyzes the degree of respondents' knowledge on specific topics
• Opinions or Judgments: such as perceived service quality
• Attitudes or predispositions of mind: We are looking for what is thought in relation to something
• Motives or explanations for specific behaviors: the question is, why certain views or acts?
• Possible future behavior: it may ask whether or not to consume a prepared product.

Concept and Structure of Questionnaires

The questionnaire is a way... Continue reading "Effective Questionnaire Design for Market Research" »