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

Key Statistical Concepts

Variables

Categorical (Qualitative) Variables: Represent characteristics or qualities. Examples include race and gender.

Quantitative Variables: Represent numerical values that differ in magnitude.

Scales of Measurement

• Nominal: Unordered categories.
• Ordinal: Ordered categories.
• Interval: Equal intervals between values.

Data Visualization

Frequency: Displays the possible values of a variable and the number of times each occurs.

Histogram: A bar graph of frequencies or percentages. Shapes can be bell-shaped, skewed, or bimodal.

Box Plot: Displays the distribution of data, including median, quartiles, and outliers.

Scatter Plot: Shows the relationship between two variables.

Distribution

• For symmetric distributions, the mean equals

• Rating Alphabetical: Documents are classified based on the name of the person or the reason for the company shown in the document.
• Geographical Breakdown: Documents are classified by place of residence of the person or entity, as the province-city-street and finally the name.
• Classification of Materials: Documents are grouped according to the topic at hand, understood as a common feature field.
• Alphabetical Materials as Logical Categories:

a) Number: Each field is assigned a number, and each subdivision a number.
b) Alphanumeric: Each field is assigned a letter or number, and subdivisions another letter or number that is added to the first component.

• Rating Chronological: Ordering progressively, by date of entry or maturity.
• Decimal Classification:

Cutting speed depends on mining velocity. The cutter's speed must remain constant throughout its journey through the ore flow.

Sampling Methods

Sampling can be performed using manual or mechanical methods. Manual methods include cone and quartering, sampling with scoops or shovels, and using sampling tubes.

Manual Sampling Techniques

Cone and Quartering

1. Stack material into a conical pile by shoveling.
2. Form a cake or 'qeqe'.
3. Divide the cake into four equal fractions.
4. Take two opposite quarters as the sample; reject the other two.
5. If the sample size is adequate, combine the two retained quarters and reduce their size.
6. Repeat steps 1-4 as necessary until the desired sample size is achieved.

Sampling Scoop or Shovel

This procedure is used when ore is moved...

Understanding Quartiles

Quartiles divide a set of ordered data into four groups with equal numbers of values. The three dividing points are Q1, the median (Q2), and Q3.

• Interquartile Range (IQR): Defined as Q3 – Q1, this represents the range of the middle half of the data. It provides a measure of spread by showing how closely the data are clustered around the median.
• Semi-interquartile Range: One half of the interquartile range.

Quartile Formulas

• Q2 (Median): (n+1) / 2
• Q1: (n+1) / 4
• Q3: 3(n+1) / 4

Outlier Formula

To identify outliers, use the following boundaries:

• Left Boundary: Q1 - 1.5(IQR)
• Right Boundary: Q3 + 1.5(IQR)

Deviation and Standard Deviation

Deviation tells you how far a single data value is from the mean (the difference between a value...

Investment Analysis II: Key Cash Flow Components

Financial leverage is debt. "Unleveraged" Free Cash Flow (FCF) means we calculate FCF without regard to how the firm is financed. Ultimately, we are valuing the assets of the project. We achieve this by estimating how much cash is generated by the asset side of the balance sheet.

Capital Expenditure (CAPEX)

Capital Expenditure (CAPEX) is defined as the original cost of investment in property, plant, or equipment and other long-term assets.

Formula for CAPEX

NET PPE (current) - NET PPE (prior) + Depreciation

Depreciation and Asset Valuation

Depreciation: Depreciation charges are intended to represent the cost of wear and tear over the asset's life. The Straight-Line Depreciation method divides the asset’s...

Understanding Accounting and Financial Statements

Accounting, within economics, is the study of business assets, the rules, and the scientific basis for the registration of economic information within an enterprise.

Key Components of Annual Accounts

Annual accounts, comprising the balance sheet, profit and loss account, the statement of changes in equity, and the memory, form a unified set. These accounts are financial statements that report on the company's results and its financial position. They include the following:

• The Balance Sheet

  Reports on the financial situation of the company at the close of a given fiscal year.

• The Profit and Loss Account

  Reports on the performance of the company for a year from its business operations.

• The Statement of

Survival Analysis Fundamentals

Understanding Survival Analysis

The main objective of survival analysis techniques is to determine differences between two or more treatments applied to a set of individuals. Each individual receives a particular treatment, and the effect (response) is measured by the occurrence of a specific event of interest (e.g., default) and the time elapsed from the start of observation until the aforementioned event occurs. Survival analysis techniques apply to data with the following characteristics:

• The dependent variable (or response variable) is the time that elapses until the individual experiences a specific event of interest, often termed death. Therefore, while the individual does not experience the event of interest,

Substance Concentration Data Grouping

The content of a substance in a liquid is given with an accuracy of 5 milligrams per liter. As such, the data can take values from this sequence:

• 120
• 125
• 130
• 135
• 140
• 145

Grouping Amplitude and Frequency Table Construction

The problem states a data set ranging between 110 and 245. However, the provided calculation for the number of steps, (247.7 - 107.5) / 25 = 5.6, implies a wider range. This calculation indicates that 6 steps (intervals) of 25 units each are needed. Since 6 intervals of 25 units cover 150 units, the apparent limits can be started with a round number to accommodate the required range.

Digital Differential Analyzer (DDA) Algorithm

• Input: The starting point (x1, y1) and ending point (x2, y2).
• Calculate:
  • dx = x2 - x1
  • dy = y2 - y1
• Find:
  • Steps = max(|dx|, |dy|)
• Compute:
  • xinc = dx / Steps
  • yinc = dy / Steps
• Plot: The initial point.
• Iterate: Add xinc and yinc repeatedly until the endpoint is reached.

Bresenham's Line Drawing Algorithm

• Calculate dx and dy.
• Initialize the decision parameter.
• Plot the starting pixel.
• According to the decision parameter, choose the next pixel.
• Update the decision parameter.
• Repeat until the endpoint is reached.

Midpoint Circle Drawing Algorithm

• If P < 0, choose the East pixel.
• If P ≥ 0, choose the South-East pixel.
• Update the decision parameter.
• Continue until x > y.
• Use 8-way symmetry to complete the circle.

Sutherland-

Central Limit Theorem (CLT)

Check the sample size n. If n is 30 or more, the sampling distribution is normal. The mean of the sample mean equals the population mean. The standard error is calculated as σ divided by the square root of n.

Z Distribution for Population Mean

• Identify values: Sample mean, population mean, σ, and n.
• Compute standard error: SE = σ / √n.
• Compute z: z = (sample mean - population mean) / standard error.
• Use the z-table to find the probability or critical value.
• Confidence interval: Sample mean ± (z-critical × standard error).

T Distribution for Population Mean

• Find the sample mean and sample standard deviation (s).
• Standard error: SE = s / √n.
• Determine the degrees of freedom (df).
• Compute t: t = (sample mean - population