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

Random Variables

A random variable is a numerical quantity that takes on different values depending on chance. There are two primary types:

• Discrete (PMF): A countable set of possible outcomes (e.g., the number of cases in an SRS from the population).
• Continuous (PDF): An unbroken continuum of possible outcomes (e.g., the average weight of an SRS of newborns selected from the population).

Key Statistical Definitions

• Population: The set of all possible values for a random variable.
• Event: An outcome or a set of outcomes.
• Probability: The proportion of times an event is expected to occur in the population.

Note: Ideas about probability are founded on relative frequencies (proportions) in populations.

Probability Calculation Example

In a given year, there... Continue reading "Understanding Random Variables and Probability Distributions" »

AdaBoost: Adaptive Boosting Explained

AdaBoost is one of the simplest and earliest boosting algorithms. The main idea behind AdaBoost is to combine many weak learners (models that do slightly better than random guessing) into one strong learner.

It works by training multiple models one after another. After each model, the algorithm checks which data points were predicted wrong. It then gives more importance (weight) to those wrongly predicted samples so that the next model focuses more on correcting those mistakes.

Each new model tries to fix the errors made by the previous ones. At the end, all models are combined using weighted voting to make the final prediction. This helps improve accuracy and reduces errors.

Key Characteristics of AdaBoost

• Combines

TDS Return Filing Due Dates

A Tax Deducted at Source (TDS) return is a quarterly statement submitted to the Income Tax Department. It is mandatory for deductors to submit these returns on time to avoid late fees, which accrue at ₹200 per day under Section 234E.

The financial year is divided into four quarters. The due date is generally the last day of the month following the end of the quarter, with an exception for the final quarter.

Differences Between PAN and TAN

While both are unique 10-... Continue reading "TDS Return Due Dates and PAN vs TAN Comparison" »

I. Algebra

1. Quadratic Equations

• Standard form: f(x) = ax² + bx + c
• Vertex form: f(x) = a(x - h)² + k (Vertex = (h, k))
• Example: f(x) = x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or 3

2. Solving Equations

• Example: (2x/3) + 1 = 5/6
  Subtract 1: 2x/3 = -1/6
  Multiply by 3: 2x = -1/2
  Divide: x = -1/4

3. Function Composition

• (f ∘ g)(x) = f(g(x))
• Example: f(x) = x² + 2x, g(x) = 3x - 1
  (f ∘ g)(2) = f(5) = 25 + 10 = 35

4. Absolute Value

• General form: f(x) = a|x - h| + k
  a = vertical stretch/shrink, h = horizontal shift, k = vertical shift
• Example: g(x) = 2|x - 3| - 1 (Stretch by 2, right 3, down 1)

II. Geometry

5. Parallel Lines & Angles

• Alternate interior angles are equal.
• Example: 5x = 70 → x = 14

6. Pythagorean Theorem

• a² + b² = c²
• Example: leg =

Fundamental Auction Concepts

Payoff: A bidder's payoff is their valuation for the item minus the price paid.

Social Surplus: This is the sum of the surpluses of all participants. The formula is: Seller's Surplus (p) + Winner's Surplus (v - p) + Loser's Surplus (0). Here, v is the winner's valuation and p is the price paid. Social surplus is maximized, and the auction is considered efficient, if the winner is the bidder with the highest valuation.

Types of Auctions

English Auction

This is a type of ascending auction where an auctioneer announces prices, and bidders accept or reject them.

• Winner: The last remaining bidder.
• Price: The second-highest price or bid.
• Information Revealed: The auctioneer learns the valuations of all bidders except for the

Floating Point Systems & Numerical Error

A Floating Point (FP) System represents numbers as: x = ± (d₀ + d₁/β + d₂/β² + ... + d_t-1/β^(t-1)). The Unit Roundoff (u) is defined as ε_machine/2, where fl(1 + ε) > 1.

Rounding to Nearest

When rounding to the nearest representable number, fl(x) = x(1 + ε) where |ε|.

IEEE 754 Standard for Floating Point

Normalized Numbers

If the exponent (e) is not equal to 0, it's a normalized FP number. The value is x = (-1)^sign ⋅ β^{(e - offset)} ⋅ (1.d₁ d₂...d_t-1).

Denormalized Numbers

If the exponent (e) is 0, the number is denormalized. The value is x = (-1)^sign ⋅ β^{(e - offset + 1)} ⋅ (0.d₁ d₂...d_t-1). The sticky bit 0 is free because it is always determined by the value of exponent e.

Exceptional Values

• If

Safety and Discipline Committees

Threat Assessment Team

TEC §37.115

Required Members:

• Principal / Assistant Principal
• Counselor
• Local Law Enforcement / School Resource Officer (SRO)

Key Responsibilities:

• Evaluates behavioral threats.
• Implements safety interventions.
• Must be trained via the Texas School Safety Center (TxSSC).

1A Panther Strategy: Cross-train your small team during the August Professional Development (PD) week so you are compliant before day one.

Placement Review Committee

TEC §37.003

Required Members:

• Two Elected Classroom Teachers
• One Alternate Teacher
• One Principal Appointee

Key Responsibilities:

• Rules on student placement if a teacher permanently removes a disruptive student from class.

1A Panther Strategy: Hold this campus election during... Continue reading "Essential Texas School Committee Compliance Strategies" »

Map Generalization

Types of Symbols

• Line Symbols: Represent real-life objects with a linear path.
• Point Symbols: Represent objects occurring at a single point on Earth's surface using a dot.
• Area (Polygon) Symbols: Represent real-life objects spread over Earth's surface using geometric shapes.

Generalization Techniques

Reality contains too much information for a single 2D map. Generalized geometry and content make a map useful. A good map suppresses less important information to highlight what needs to be seen.

• Selection: Only relevant line, point, and area features are chosen.
• Classification: Grouping similar features and using a common symbol to represent them.
• Simplification: Reduction of unnecessary detail.
• Smoothing: Smoothing out abruptly joined

Boolean Algebra Fundamentals

Formulating Expressions: SOP and POS

Sum of Products (SOP) Formulation Steps

1. Circle rows in the truth table where the output $Y = 1$.
2. Identify the minterms corresponding to the circled rows.
3. Sum (OR) the selected minterms to form the final expression.

Product of Sums (POS) Formulation Steps

1. Circle rows in the truth table where the output $Y = 0$.
2. Identify the maxterms corresponding to the circled rows.
3. Multiply (AND) the selected maxterms to form the final expression.

Order of Evaluation in Boolean Algebra

Operations are evaluated in the following sequence:

1. Parentheses
2. NOT (Complementation)
3. AND (Multiplication)
4. OR (Addition)

Fundamental Boolean Laws and Theorems

• Identity Laws

  • $A + 0 = A$
  • $A \cdot 1 = A$

• Null Laws

  • $A + 1 = 1$
  • $A \cdot

Linear Algebra and Matrix Properties

1. State one property of a square matrix that will guarantee it is singular.
   There are many possibilities, some of which are: having determinant 0, having dependent columns, having dependent rows, and having a non-trivial null-space.
2. Can a system of linear equations have exactly 2 solutions?
   No, a system of linear equations can have either 0, 1, or infinitely many solutions.
3. What advantages does the Cholesky factorization have over Gaussian elimination?
   The Cholesky factorization only requires about half the operations of Gaussian elimination, and pivoting is never required for stability.
4. Can an n × n positive-definite matrix be singular? What about an n × n symmetric positive-definite matrix?
   The condition