Notes, summaries, assignments, exams, and exercises for Mathematics for Other courses

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...

Understanding Kurtosis: Distribution Shape

Kurtosis is a statistical measure that describes the shape of a distribution’s tails compared to a normal distribution. It tells us whether the data are heavy-tailed or light-tailed.

In simple terms, kurtosis indicates the degree of peakedness and the presence of outliers in data.

Types of Kurtosis

• Mesokurtic: Normal distribution (kurtosis = 3).
• Leptokurtic: More peaked, heavy tails (kurtosis > 3).
• Platykurtic: Flatter peak, light tails (kurtosis < 3).

Key Concepts in Hypothesis Testing

1. Null Hypothesis (H₀)

It is a statistical statement that assumes no effect or no difference.

Example: “There is no difference between two groups.”

2. Alternative Hypothesis (H₁ / Hₐ)

It is the opposite of the...

Key Advantages of Bonds for Investors

Investing in bonds offers several key benefits:

• Source of Current Income: They are a good source of regular income.
• Relative Safety: Investment in bonds is relatively safe from large losses.
• Priority in Default: In case of default, bondholders receive their payments before shareholders can be compensated.

Comprehensive Bond Classification

Bonds are classified by their key features, which include:

• Form of Payment
• Coupon Payment
• Collateral
• Type of Circulation
• Type of Issuers
• Recall Possibility
• Place of Circulation
• Quality
• Other Miscellaneous Types

By Form of Payment

• Non-interest-bearing Bonds
• Regular Serial Bonds
• Deferred-interest Bonds
• Income Bonds
• Indexed Bonds
• Optional Payment Bonds

By Coupon Payment

• Coupon Bonds
• Zero-coupon Bonds
• Full

Fundamental Accounting Principles and Practices

Core Accounting Definitions

Assets = Liabilities + Owner’s Equity: This is the fundamental accounting equation, representing the balance of a company's financial position.

Double-Entry Accounting

The recording of both debit and credit parts of every transaction, ensuring the accounting equation remains balanced.

Objective Evidence Principle

Requires that a source document (e.g., invoice, receipt) be prepared for every entry in a journal, providing verifiable proof of a transaction.

File Maintenance

The process of arranging accounts in a general ledger, assigning account numbers, and keeping records current.

Opening an Account

Writing an account title and number on the heading of an account in the ledger....

Operating Segments & Disclosures

Revenue Test Criteria

Both external and intersegment revenue are included in the revenue test.

Profit & Loss (P&L) Test

The P&L test includes 10% of the higher absolute value of intersegment revenue and expenses.

Management's Business Evaluation

Management evaluates business results along product, customer, or geographic lines, not based on legal entity results.

Defining an Operating Segment

• Includes state
• Reviewed by the chief decision maker
• Discrete information available
• Product is the primary breakdown, with geographic and customer as secondary.

Operating Segment Disclosure Rules

• Must reconcile to GAAP.
• For immaterial operating segments, revenue, P&L, and assets must be disclosed.

Non-Operating Segment

Core Machine Learning Definitions

a) Define Machine Learning: Machine Learning is a subset of AI where a system learns patterns from data and makes decisions or predictions without being explicitly programmed for every task.

b) What is over-fitting in Machine Learning?: Over-fitting is when a model learns the training data too well, including noise. It performs very well on training data but poorly on new unseen data.

c) What is underfitting?: Underfitting is when a model is too simple to capture the underlying pattern in the data. It performs poorly on both training and test data.

d) Define bias and variance: Bias is error due to oversimplified assumptions in the model—leads to underfitting. Variance is error due to the model being too sensitive...

Financial Valuation Metrics

Market Capitalization = Market value of equity = Shares × Price of share

Enterprise Value = Market Value of Equity + Debt − Cash

EBITDA and EBIT Formulas

• EBITDA = Net Income + DA, Taxes, and Interest
• EBITDA = Revenue − Expenses (excluding DA, Taxes, and Interest)
• EBITDA = EBIT + Depreciation + Amortization
• EBIT = Revenues − Costs − Depreciation

Income and Cash Flow Calculations

Income Tax = EBIT × τ

Unlevered Net Income = (Revenues − Costs − Depreciation) × (1 − τ)

Free Cash Flow = (EBIT) × (1 − τ) + Depreciation − CapEx +/− ΔNWC

Investment Decision Rules

Sunk costs are costs that have been or will be paid regardless of the decision to undertake an investment.

NPV = PV(Benefits) − PV(Costs). Invest...

Agent Environments and PEAS

PEAS: Performance, Environment, Actuators, Sensors

Environment Properties

• Fully vs. Partially Observable: Full information or hidden information?
• Deterministic vs. Stochastic: Outcome is known or probability is known (if probability is unknown, it is Nondeterministic).
• Episodic vs. Sequential: Independent actions vs. sequential (dependent) actions.
• Static vs. Dynamic: No environment change vs. environment change over time.
• Semidynamic: The environment does not change with time, but the agent's performance score does.
• Discrete vs. Continuous: Distinct chunks (e.g., Chess) or smooth/measurable values (e.g., Driving).
• Known vs. Unknown: Are the rules of the world given or must they be learned?
• Single vs. Multi-agent: One agent

Essential Statistical & Numerical Concepts

This document covers fundamental questions and answers across various topics in statistics, numerical methods, and data analysis. Each section provides a concise explanation of key concepts.

1. What is Interpolation?

Interpolation is a mathematical technique used to estimate unknown values that fall between known data points. It involves creating a function or model based on the given data and using it to predict values within the range of the data, rather than extrapolating outside it. This method is commonly applied in fields like data analysis, engineering, and computer graphics for filling in missing data or smoothing curves.

2. Regula Falsi Method Formula for Finding Roots

The formula for finding...

Essential Financial Statement Formulas

• Stockholder Equity: Total Assets – Total Liabilities
• Retained Earnings: Net Income – Dividends Declared
• Net Income: Sales Revenue – Expenses
• Gross Profit: Sales Revenue – Cost of Goods Sold (COGS)
• Cost of Goods Sold (COGS) based on Rate: (1 – Gross Profit Rate) × Net Sales

Inventory Accounting Adjustments (LIFO and FIFO)

• Calculating LIFO Reserve: FIFO Ending Inventory Cost – LIFO Ending Inventory Cost
• Adjusting Balance Sheet from LIFO to FIFO Inventory: LIFO Reserve + LIFO Inventory

Accounting Estimates and Depreciation

Changing Accounting Estimates

• Book Value at Date of Change: Costs – Accumulated Depreciation
• New Remaining Useful Life: Original Life – Years Depreciated + Additional Years
• Depreciation