Notes, summaries, assignments, exams, and exercises for Mathematics for Secondary education

Econometric Notation: Residuals and Estimates

Residuals are denoted by e (Sample Regression Function, SRF) and the true error term by u (Population Regression Function, PRF).

• Coefficients: Estimated coefficients (e.g., $\hat{\beta}_0$ and $\hat{\beta}_1$) in the SRF are denoted with hats (caps). True coefficients ($\beta_0, \beta_1$) in the PRF are without hats.
• Dependent Variable: The predicted value $\hat{Y}_t$ (the value derived from the regression equation without the residual) is denoted with a hat. The actual observed value $Y_t$ (which includes the residual or error term) is without a hat.

Fundamentals of Hypothesis Testing

Hypothesis testing makes inferences about the validity of specific economic (or other) theories from a sample of the...

Different stages of Spain in the XIX

Financial Market Scenarios: Fixed Income and Derivatives

Sildavia: Term Structure and Yield Curve Construction

In Sildavia's financial market, we assume the existence of a series of zero-coupon bonds and conventional bonds. Each bond type presents specific traits:

• Zero-Coupon Bonds: Defined by spot rates and coupon interest rates.
• Conventional Bonds: Defined by specific characteristics (data assumed to be provided in an accompanying table).

Based on the provided data, the objective is to construct the TSIR (Term Structure of Interest Rates) and the Yield Curve for both kinds of bonds.

Moravia: Calculating Forward Rates

Considering the TSIR shown in Table 1, calculate the corresponding forward rates (r_t).

Furthermore, if the market involves an inverted

Food and Nutrition

Food is the act of providing sustenance to the human body, while nutrition encompasses the physiological processes by which the body receives, transforms, and utilizes the chemical components in food.

Inherited Diseases

• Hemophilia
• Huntington's disease
• Cystic fibrosis
• Color blindness
• Phenylketonuria

Measures of Dispersion

Measures of dispersion indicate how close the data are to the average.

Vaccines

Vaccines are preparations containing killed or attenuated microorganisms. They are introduced into our bodies to produce antibodies that kill the organism, providing immunity.

Common Pathogens

Pathogens such as bacteria and fungi are the most common causes of infectious diseases.

Health and Disease

Health is a state of complete physical, mental,...

1. Simplifying Algebraic Expressions

How to Simplify Algebraic Expressions

• Multiply numbers and letters together.
• Combine like terms (terms with the same letters and powers).

Example:
−2ac × 4bd = −8abcd

Example:
5ab − 8b²a + ba = 5ab − 8ab² + ab = −8ab² + 6ab

2. Expanding Brackets (Distributive Law)

How to Expand Brackets

• Multiply everything inside the bracket by what is outside.

Example:
−4(x + 7) = −4x − 28

Example:
5(x − 3) + 2(6 − x) = 5x − 15 + 12 − 2x = 3x − 3

3. Solving Equations

How to Solve Equations

• Get all variables (e.g., x's) on one side and numbers on the other.
• Perform the same operation on both sides of the equation.

Example:
3x + 2 = 17
3x = 15
x = 5

Example:
9x − 8 = 4x + 7
5x = 15
x = 3

4. Solving Inequalities

How

Text Connectors: Uniting Ideas

Connectors serve to unite the parts of a text, ensuring coherence and flow. They can be categorized by their function:

Types of Connectors

• Time Connectors: Used to organize events in a text following a temporal order (e.g., first, then, next, finally).
• Order Connectors: Used to organize the different parts of a text from a logical standpoint (e.g., firstly, secondly, moreover, in conclusion).

Punctuation Marks: Guiding Your Reader

Punctuation marks are essential for clarity and meaning in written language.

Question Marks (?)

Used at the beginning and end of direct questions.

Exclamation Marks (!)

Used at the beginning and end of statements that express strong feelings, surprise, or emphasis.

Note: Punctuation marks are not

Adjusting Events

The following events require adjustments in financial statements:

• Settlement of a court case
• Asset impairments (asset not well accounted for)
• Determination of costs of assets purchased or sold before the reporting period
• Determination of profit or bonus payment
• Fraud or errors

Non-Adjusting Events

The following events do not require adjustments in financial statements:

• Major business combinations or disposal of a subsidiary (sell of a subsidiary)
• Plan to discontinue operations
• Purchases of assets or expropriation by government
• Destruction of a plant (by a fire, earthquake…)
• Restructuring
• Ordinary shares transactions
• Changes in asset prices or foreign exchange rates after reporting period
• Changes in tax rates or tax law
• New commitments or

Problem 4: Bank Savings Distribution

The average savings in the clients' accounts of a certain bank follow a normal distribution N(50,000, 15,000) (mean = 50,000 €, standard deviation = 15,000 €). The bank distributes its clients into three categories:

• Category A: Those clients that have less than 20,000 € in their savings account
• Category B: Those clients that have between 20,000 € and 80,000 € in their savings account
• Category C: Those clients that have more than 80,000 € in their savings account

Calculate:

4a) What percentage of clients belong to Category A?

Use the standard normal variable Z = (X - μ) / σ.

For X = 20,000 €:

Z = (20,000 - 50,000) / 15,000 = -30,000 / 15,000 = -2.000

The cumulative probability Φ(-2.00) ≈ 0.0228,...

Parity and Set Theory Proofs

1. Direct proof: odd + even is odd

Statement: If x is an odd integer and y is an even integer, then x + y is odd.

Proof: Suppose x = 2k + 1 and y = 2l for integers k, l. Then
x + y = 2k + 1 + 2l = 2(k + l) + 1, which is of the form 2·(integer) + 1, hence odd. ∎

2. Contrapositive proof: even implies next is odd

Theorem: If n is an even integer, then n + 1 is odd.

Proof by contraposition: The contrapositive is: If n + 1 is not odd (i.e., even), then n is not even (i.e., odd). Suppose n + 1 is even. Then n + 1 = 2k for some integer k. Thus n = 2k − 1 = 2(k − 1) + 1, which is odd. Therefore the contrapositive is true, and so the original implication holds. ∎

3. Proof by contradiction: even implies next is odd

Theorem:

Adverbs

An adverb is a word that acts as the core of an adverbial phrase. It modifies a verb, an adjective, or another adverb.

Classification of Adverbs

• Place: out there, here, up close, in front, within, above, across, away, etc.
• Time: now, before, even yesterday, after, today, then, tomorrow, never, soon, always, yet, etc.
• Quantity: something, pretty, almost, too much, well, nothing, just, so, so much.
• Manner: agile, well, good, bad, fair, bravely, and so on.
• Order: before, after, later, first, respectively, and so on.
• Affirmation: true, certainly, of course, actually, probably, yes, indeed, and so on.
• Negation: never, no, never, no, etc.
• Doubt: perhaps, possibly, probably, maybe, perhaps, etc.
• Question: how, when, where.
• Relative: how, when, where.

Prepositions

A...