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

Calculating the Future Value of an Annuity Due

Step 1: Determine the Variables

The problem provides the following details:

• Annual payment: Rs. 200. Therefore, the half-yearly payment (Pmt) is:
  
  Rs. 200 / 2 = Rs. 100
  Rs. 200 / 2 = Rs. 100
• Annual interest rate (r): 4% or 0.04. Since the interest is compounded half-yearly, the interest rate per period (i) is:
  
  0.04 / 2 = 0.02
  0.04 / 2 = 0.02
• Term: 20 years. Payments are made half-yearly, so the total number of periods (n) is:
  
  20 × 2 = 40
  20 × 2 = 40
• The annuity type is an annuity due, meaning payments are made at the beginning of each period.

Step 2: Apply the Future Value Formula

The formula for the Future Value (FV) of an annuity due is given by:

FV = Pmt × [((1 + i)^n - 1) / i] × (1 + i)

FV = Pmt × [((1

Customer Value: (Quality, Time, Flexibility, Customer Experience, Innovation) / Price

Sustainability Paradigms: Economic (Viable, Equitable), Environment (Viable, Bearable), Social (Bearable, Equitable)

- Efficiency (Optimization), Differentiator (Innovation), Driver (Motivation) | - Audits (Assess Sustainability Performance)

Table 7.3: Factors for Calculating Three-Sigma Limits for the X (Bar) Chart and R-Chart

*A sample is out of control if its value falls below the LCL or above the UCL*

lim x->0 sinx/x = 1 | H.A.: compare degrees | V.A.: denom = 0 | Continuous if: f(a), lim, equal

DERIVATIVES: (x^n)'=nx^(n-1), (e^x)'=e^x, (a^x)'=a^x ln a, (lnx)'=1/x | (uv)'=u'v+uv', (u/v)'=(u'v-uv')/v^2 |chain: (f(g(x)))'=f'(g(x))g'(x)

TRIG DERIVATIVES: (sin)'=cos, (cos)'=-sin, (tan)'=sec^2 | (sec)'=sec·tan, (csc)'=-csc·cot, (cot)'=-csc^2

CRITICAL POINTS:f'=0 or DNE ⇒ crit pt | f'>0 inc | f'<0 dec | f''>0 conc up | f''<0 conc down | inflec = f'' signchange

INTEGRATION: ∫x^n dx = x^(n+1)/(n+1)+C | ∫e^x dx = e^x+C | ∫a^x dx = a^x/ln a+C | ∫1/x dx = ln|x|+C | ∫sin x dx

= -cos x+C | ∫cos x dx = sin x+C

FTC: Part 1: d/dx ∫_a^x f(t) dt = f(x) | Part 2: ∫_a^b f(x) dx = F(b)-F(a)

AREA & VOLUME: A = ∫_a^b (top - bot)... Continue reading "5 hr 20 min 20 sec corresponds to a longitude difference of" »

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(

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" »

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)

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 -