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

Section A: R Basics and Data Types (Weeks 1-4)

Model Questions and Answers

Q: Create a vector v (3, NA, Inf, -Inf). Explain adding 5 to v.

A: The operation is element-wise. Missing values (NA) propagate, resulting in NA. Infinite values (Inf, -Inf) remain infinite.

v <- c(3, NA, Inf, -Inf)
print(v + 5)
# Output: [1]  8 NA Inf -Inf

Q: Given vector a, write code to count and replace NAs.

A: Assuming a <- c(10, 15, NA, 20).

• Count NAs: sum(is.na(a)) → 1.
• Replace NAs (e.g., with 0): a[is.na(a)] <- 0.

Q: Explain the difference between a[a > 12] and a[which(a > 12)].

A: Both select elements greater than 12 (15, 20). However:

• a[a > 12] → [1] 15 NA (Uses logical indexing; preserves the position of NA in the original vector as NA).
• a[which(

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:

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

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;

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

Understanding Inheritance in Python

Inheritance allows a class to use and extend the properties and methods of another class. It promotes code reusability and reduces duplication. Think of it as “child learns from parent.” 👨‍👦‍💻

Code Duplication: Program Without Inheritance

When classes share common attributes or methods (such as first_name, last_name, and get_age), implementing them separately leads to redundant code, as shown below:

import datetime

class TennisPlayer:
    def __init__(self, fname, lname, birth_year):
        self.first_name = fname
        self.last_name = lname
        self.birth_year = birth_year
        self.aces = []

    def get_age(self):
        now = datetime.datetime.now()
        return now.year -