Notes, summaries, assignments, exams, and exercises for Mathematics for Baccalaureate

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

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

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

Bolzano's Theorem

If f(x) is continuous on [a, b], there exists at least one value c in (a, b) such that f(c) = 0.

Consequences

If f(a) > g(a) and f(b) < g(b), there exists c in (a, b) such that f(c) = g(c).

Mean Value Theorem

If f(x) is continuously differentiable on [a, b], then there exists a point where the derivative f'(c) equals the average rate of change.

Linear Combinations and Geometry

A linear combination is defined as: a(x₁, y₁, z₁) + b(x₂, y₂, z₂) + c(x₃, y₃, z₃) = (x₄, y₄, z₄).

Distance Formulas

• Between two points: The magnitude of the vector.
• Point to line: dist(P, r) = |AP × Vd| / |Vd|, where A is a point on the line and Vd is the direction vector.
• Point to plane: |ax + by + cz + d| / √(a² + b² + c²).

Working Capital

Working Capital measures the capacity for payment in the ordinary course of business activity. It's calculated as: Current Assets (CA) - Current Liabilities (CL)

• CA > CL: Positive Working Capital. The business has the potential for investment. Working Capital should never exceed 10% of CA, as these are idle funds.
• CA < CL: Negative Working Capital. This may indicate a suspension of payments or insolvency. It usually signifies mismanagement in the negotiation of ordinary business activity, but it doesn't always mean a bad situation.

Acid Test

The Acid Test measures a company's capacity to meet all of its short-term debts. It's calculated as: (Current Liabilities - Treasury) / Available. This indicates immediate liquidity;...

Question 6: Vector Spaces and Subspaces

Finding the Zero Vector

1. Take the general scalar multiplication formula r ◦ (a, b, c, d).
2. Plug in r = 0.
3. Read off the resulting matrix; that is the zero vector. (Note: 0 ◦ anything = the zero vector in any vector space.)

Proving a Subset U is a Subspace

1. Identify which entries of U are locked (fixed numbers) and which are free (variables).
2. Check if the zero vector (from part a) fits the locked entries of U.
3. Take two general elements of U (using the same locked numbers but different free letters) and combine them with addition. Check that the locked entries stay locked.
4. Take one general element of U and scale it with ◦. Check that the locked entries stay locked.
5. If all three subspace conditions hold, then U

Data Science, Machine Learning, and Artificial Intelligence

Chapter 1: Understanding Variables

Types of Variables

• Categorical: Smoker (current, former, no)
• Ordinal: Non, light, moderate, heavy smoker (ordered categories)
• Quantitative: BMI, Age, Weight (numerical measurements)

Key Definitions

• Observation: Measurements are made (individual or aggregate).
• Variable: The generic characteristic we measure (e.g., age).
• Value: A realized measurement (e.g., 27).

Chapter 2: Statistical Studies

Surveys: Census and Sampling

• Goal: Describe population characteristics.
• Census: Attempts to reach the entire population (costly, time-consuming).
• Sampling: Uses a sample of the population (allows for inferences, saves time and money).
• Simple Random Sampling: Based on probability.
• Issues with Sampling: Under-coverage, volunteer bias,

Probability Fundamentals

• Mutually Exclusive Events: If A and B are mutually exclusive, then P(A ∪ B) ≠ 0.
• Independent Events: Events are independent if their probabilities match, such as P(A) = P(B) or P(A|B) = P(A).
• Conditional Probability Example: Find P(driver is age 65+ | driver is female) = P(driver is female and 65+) / P(driver is female).
• Key Formulas:
  • P(S ∩ Y) = P(S ∩ Y) / Total
  • P(S|Y) = P(S ∩ Y) / P(Y)
  • P(Y ∪ S) = P(Y) + P(S) - P(Y ∩ S)

Probability Distributions

Expected Value Calculation

The payoff (X) for a lottery game has the following probability distribution. What is the expected value of X? (Hint: Find the value of 'a' first.)

Given values: 2 (0.4), 5 (0.3), 7 (0.2), 10 (a).
Calculation: 1 - (0.4 + 0.3 + 0.2) = 0.1, so a =...

Agile Methodology Fundamentals

The agile approach differs from classic project management because it is iterative and incremental.

Sprint Planning Objectives

The sprint planning meeting has two primary objectives: defining the functionalities to be developed in the next sprint and identifying the necessary tasks to complete them.

The Product Owner Role

The product owner is the person within the company who manages business needs and is responsible for the product backlog.

Estimated vs. Real Effort

Estimated effort refers to the projected time required, while real effort is the actual time spent on a task.

Scrum Daily Meeting

The main objective of the Scrum daily meeting is to monitor project status and make decisions to achieve the sprint goal. The...