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

Quick revision table for Mathematics and Statistics exams.

Unit I: Integration and Data Presentation

Unit II: Central Tendency and

Two-State Actuarial Model

The Two-State Model (also known as the Dead-Alive or Binary Model) is a fundamental actuarial framework used to represent processes that exist in one of two possible states, such as Alive/Dead or Working/Retired. It is widely utilized in life insurance and pension modeling to estimate the probability of transition between states. The model assumes that at any given time, an individual occupies only one state, allowing actuaries to calculate premiums, reserves, and expected present values by simplifying complex uncertainties into binary outcomes.

Core Assumptions

• Binary States: The system exists in only one of two states at any time.
• Markov Property: Transitions depend solely on the current state.
• Constant Probabilities:

Taylor Polynomials and Derivatives

Start by finding the derivatives of f(x) as needed. In the example question, it wants the degree 3, so we calculate up to the 3rd derivative, f'''(x), and then use the formula.

Solving Differential Equations

The equation ds/dt = sin(t) shows us that it is a pure time differential, meaning taking the integral of ds/dt will get us s(t). Since we know s(0) = 0, we use this to find the constant C. Once found, substitute it into s(3), and you have solved the problem.

Euler's Method for Numerical Solutions

To use Euler's Method, create your table with x moving at the rate given as Δ. Use the initial points and the formula provided. If given a function f(x), find the derivative. If already given a derivative or dy/dx,... Continue reading "Calculus Problem Solving Techniques and Formulas" »

Combine Independent Unbiased Estimators

Let d₁ and d₂ be independent unbiased estimators of θ with variances σ₁² and σ₂², respectively:

• E[d_i] = θ for i = 1,2.
• Var(d_i) = σ_i².

Any estimator of the form d = λ d₁ + (1 - λ) d₂ is also unbiased for any constant λ.

The variance (mean square error for an unbiased estimator) is
Var(d) = λ²σ₁² + (1 - λ)²σ₂².

To minimize Var(d) with respect to λ, differentiate and set to zero:

d/dλ Var(d) = 2λσ₁² - 2(1 - λ)σ₂² = 0.

Solving gives the optimal weight

λ_* = σ₂² / (σ₁² + σ₂²).

Question 1: Posterior PMF for a Third Dice Roll

Assume there are five dice with sides {4, 6, 8, 12, 20}. One of these five dice is selected uniformly at random (probability 1/5) and rolled twice. The two observed results are... Continue reading "Optimal Estimators, Dice Posterior & Statistical Problems" »

Introduction to Statistical Learning

Key Notation

• Y: Response (dependent/output) variable - quantitative or qualitative
• X₁,...,Xₚ: Predictors (features/covariates/independent variables)
• p: Number of predictors
• n: Number of observations
• i: Subject index (i = 1,...,n)
• j: Variable index (j = 1,...,p)
• Data: (Yᵢ, Xᵢ), i = 1,...,n

Types of Learning

• Supervised learning: Have both Y and X (prediction or inference) - PRIMARY FOCUS
• Unsupervised learning: Have X but no Y

Model for Data

Yᵢ = f(Xᵢ) + εᵢ, i = 1,...,n
Assumptions: E(εᵢ) = 0, var(εᵢ) = σ², εᵢ are mutually independent
Properties: E(Yᵢ|Xᵢ) = f(Xᵢ), var(Yᵢ|Xᵢ) = σ²

Prediction vs Inference

Calculating Karl Pearson Correlation Coefficient

Question 11: Calculate the coefficient of correlation from the following data:

To calculate Karl Pearson's coefficient of correlation (r), we use the formula:

r = ∑dx·dy / √(∑dx² × ∑dy²)

Where:

• dx = X - X̄
• dy = Y - Ȳ

Step 1: Find the Means

First, calculate the mean of X (X̄) and Y (Ȳ):

X̄ = (12 + 9 + 8 + 10 + 11 + 13 + 7) / 7 = 70 / 7 = 10

Ȳ = (14 + 8 + 6 + 9 + 11 + 12 + 3) / 7 = 63 / 7 = 9

Step 2: Prepare the Calculation Table

Step 3: Substitute into the Formula

Substitute the sums from the table into the correlation formula:

r = 46 / √(28 × 84)
r = 46... Continue reading "Correlation Coefficient, Regression, and Chebyshev's Theorem" »

Evaluation Metrics for ML Models

• Accuracy: The ratio of correctly predicted instances.

• Precision: Correct positive predictions divided by total predicted positives.

• Recall: Correct positive predictions divided by actual positives.

• F1 Score: The harmonic mean of precision and recall.

  

K-Nearest Neighbors (KNN) Algorithm

• A classification algorithm that works by finding the 'k' closest training examples to a data point.

• Strengths: Simple to understand, effective for smaller datasets.

• Weaknesses: Sensitive to irrelevant features and the scale of the data.

• Applications: Image recognition, recommendation systems.

Ensemble Learning Techniques

• Combines multiple models to improve predictive performance.

• Methods:

  • Bagging (e.g., Random Forests)
  • Boosting (e.g., AdaBoost)

(a) Alphabet, String, and Language Definitions

• Alphabet (Σ): A finite, non-empty set of symbols. Example: Σ = {0, 1} or Σ = {a, b}.
• String: A finite sequence of symbols chosen from an alphabet. Example: 0110, abba.
• Language (L): A set of strings over an alphabet Σ. Example: L = {a, ab, aab, aaab, ...} = a⁺b over Σ = {a, b}. ε denotes the empty string (length 0).

(b) DFA vs. NFA

• DFA: For each state and input symbol, exactly one next state is defined. No ε-transitions are allowed.
• NFA: For each state and input symbol, zero, one, or more next states are possible. ε-transitions are allowed.
• Power: Both accept the same class of languages (Regular Languages). Every NFA can be converted to an equivalent DFA.
• Design: DFA usually has more states;