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

Metallurgical sample preparation for microscopy is a crucial step in analyzing the microstructure and properties of metallic materials. Proper sample preparation is essential to obtain accurate and meaningful results. The process involves several key steps:

1. Sample Selection for Microscopy

Choose a representative portion of the material to be analyzed. Ensure that the sample is free from external contaminants and has a flat surface for preparation.

2. Precision Cutting Techniques

Use a precision cutting method to obtain a small section of the material for analysis. Common cutting techniques include abrasive cutting (using a saw with abrasive blades), wire cutting, or electrical discharge machining (EDM) for hard materials.

3. Sample Mounting for

Introduction to Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions.

Types of Statistics

• Descriptive Statistics: Methods of organizing, visualizing, and summarizing information from samples or populations.
• Inferential Statistics: Methods of using information from a sample to draw conclusions regarding the population.

Example: A survey of 2,000 students (3rd to 12th grade) found that they devoted an average of 7 hours and 38 minutes each day to using electronic media.

Key Definitions

• Data: Information coming from observations, counts, measurements, or responses.
• Population: The collection of all outcomes, measurements, or responses (sometimes called a census). A numerical description

Bond Characteristics

• Coupon: The interest payment made by the bond issuer, usually expressed as an annual percentage of the bond's face value.
• Par (Face Value): The amount the bondholder receives when the bond matures, typically $1,000.
• Term to Maturity: The time remaining until the bond's maturity date when the issuer must repay the bond's par value.
• Denomination: The face value of the bond, usually in increments of $1,000.
• Quotation: Bonds are quoted as a percentage of their face value (e.g., a bond quoted at 95 is selling for 95% of $1,000, or $950).

Bond Prices, Yield to Maturity (YTM), Current Yield, and Rate of Return (HPR)

• Bond Prices: The market price of a bond depends on interest rates. Prices and interest rates have an inverse relationship.

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

You requested the full list of major probability distributions used in computational statistics, machine learning, and data science. Below is a classification with key examples.

Types of Probability Distributions

1. Discrete Distributions (Countable Outcomes)

• Bernoulli Distribution: Binary outcome (0 or 1, e.g., a coin toss).
• Binomial Distribution: Number of successes in n independent trials.
• Negative Binomial Distribution: Number of trials required to achieve k successes.
• Geometric Distribution: Number of trials until the first success.
• Poisson Distribution: Number of events occurring in a fixed interval of time or space.
• Multinomial Distribution: Generalization of the binomial distribution for multiple categories.
• Discrete Uniform Distribution: Each

1. Sample Space: Binary Signal Example

Definition: A sample space (S) is the set of all possible outcomes of a random experiment, usually denoted by S.

Example: In digital communication, a binary signal has only two possible values: 0 and 1. Hence, the sample space is S = {0, 1}. This means every outcome of the experiment must belong to this set.

2. Event Definitions and Types

Definition: An event (E) is any subset of the sample space, representing one or more outcomes of an experiment.

Types:

• Simple Event: Contains only one outcome. Example: E = {0}
• Compound Event: Contains more than one outcome. Example: E = {0, 1}
• Impossible Event: An event that cannot occur, represented by the empty set (∅). Example: Getting a ‘2’ in a binary system.

3. Deterministic

Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

2. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0,... Continue reading "Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques" »

Why Reinforcement Learning?

Reinforcement Learning (RL) is important because it enables machines to learn optimal behaviors through interaction with their environment, without needing labeled input/output pairs. It is especially useful in scenarios where the best actions are not immediately known, such as game playing, robotics, or dynamic pricing.

In RL, the agent gradually learns to take actions that maximize cumulative future rewards. Unlike supervised learning, RL focuses on long-term outcomes, rather than just immediate correctness.

Main Elements of Reinforcement Learning

• Agent: The learner or decision-maker.
• Environment: Everything the agent interacts with.
• State (S): The current situation of the environment.
• Action (A): Choices available to

Current Ratio

The current ratio measures a company's ability to pay short-term obligations with its current assets.

• Positive (>1): Current assets can meet current liabilities. Specifically, assets cover [X]% of the liabilities.
• Negative (<1): Current assets cannot meet all current liabilities. They only cover [X]%, indicating a risk of short-term default.

Acid Test (Quick Ratio)

This metric determines if a company can meet its short-term debts without relying on inventory sales.

• Positive (0.8–1.2): With bank cash and accounts receivable, the company can satisfy [X]% of current liabilities without selling stock.
• Negative (<0.8): The company can only satisfy [X]% of liabilities without stock, making it too dependent on inventory sales to

Inventory Fundamentals

One use of inventory is to provide a hedge against inflation. ABC analysis divides an organization's on-hand inventory into three classes based upon annual dollar volume. Cycle counting is a process by which inventory records are verified. The difference(s) between the basic EOQ model and the production order quantity model is that the production order quantity model does not require the assumption of instantaneous delivery. Extra units that are held in inventory to reduce stockouts are called safety stock. Inventory record accuracy would be decreased by increasing stockroom accessibility. The two most important inventory-based questions answered by the typical inventory model are when to place an order and how many of