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

Core Probability Definitions and Formulas

• Independence: Events A and B are independent if P(A ∩ B) = P(A)P(B).
• Mutual Exclusivity: Events A and B are mutually exclusive if P(A ∩ B) = 0.

Key Relationships

• If P(A) + P(B) > 1, then A and B are not mutually exclusive. (True)
• If P(A) + P(B) = 1, A and B are not necessarily mutually exclusive. (False, unless A and B are complements)

Conditional Probability and Addition Rule

• Multiplication Rule: P(A ∩ B) = P(A) P(B|A).
• Conditional Probability Definition: P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0.
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Advanced Probability Relationships

Bayes' Theorem Setup

Given P(A|B) = p, P(B|A) = q, and P(A) = r. Using the definition of conditional probability:... Continue reading "Conditional Probability and Discrete Distributions" »

Limits of Sequences

We will often need to specify that a number x is close to another number a. However, this doesn’t mean anything unless we specify how close. If ε is a positive number, then the statement “x is within ε of a” does have meaning. It means that the distance between x and a is less than ε – that is, |x − a| ε.

Theorem 2.1.1

If x, y, a, and ε are real numbers with ε > 0, then (a) |y|.

Theorem 2.1.2 (Triangle Inequality)

If a and b are real numbers, then (a) |a + b| ≤ |a| + |b|; and (b) ||a| − |b| ≤ |a − b|. Proof: For part (a), we observe that −|a| ≤ a ≤ |a| and −|b| ≤ b ≤ |b|. If we add these inequalities, the result is
−(|a| + |b|) ≤ a + b ≤ |a| + |b|. By the preceding theorem (with “&... Continue reading "Understanding Limits of Sequences in Mathematics" »

Geometry

Angles

Adjacent angles - that have a common vertex and side, but no common interior points

Angle - formed by two rays with a common endpoint called the vertex

Complementary angles - whose measures add up to 90°

Congruent angles - that have the same measure

Supplementary angles - two angles whose measures add up to 180°

Vertical angles - the nonadjacent angles formed by two intersecting lines

Lines

Midpoint - the point that divides a segment into two congruent segments

Parallel lines - in a plane that never meet

Perpendicular lines - that intersect at 90° angles

Transversal - a line that intersects any two or more other lines

Triangles

Triangle Inequality Theorem - the sum of the lengths of any two sides of a triangle is greater than the length... Continue reading "Essential Geometry and Algebra Glossary" »

Statistics: Key Concepts and Methods

1. Statistics is a collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions.

2. Statistical methods consist of a series of procedures for managing, analyzing, and collecting qualitative and quantitative research data.

3. A population includes all objects of interest, whereas a sample is only a portion of the population.

Example: The population could be all students of a postgraduate program, and a representative sample of this population could be 400 students from different studies of the program.

Understanding Variables in Statistics

4. A variable is any characteristic, number, or quantity that can be measured... Continue reading "Key Concepts and Methods in Statistical Analysis" »

Unit 6: Costing and Production Systems

1. Production Systems

The two common cost accumulation systems are job order costing and process costing. The election to use one or the other of these systems depends primarily on the nature of the product and the manufacturing process by which it is created. Process production and job-order production are the extreme possibilities when considering the heterogeneity of products. The difference between order accounting and section accounting is the aim of the cost system, and the difference between process costing and job costing is the method to assign the cost to products.

1.1. Job Order Costing System

Job order costing suits the processes where it is necessary to assign costs to a specific amount of production... Continue reading "Costing and Production Systems: Job Order and Process Costing" »

Total Cost = w₁x₁ + w₂x₂ + … + w_nx_n

Isocost function:

x₂ = -w₁x₁/w₂ + tc/w₂ where the slope is: -w₁/w₂ ratio of relative factor prices with negative sign

Cost minimization mate: -w₁/w₂ = -MP₁/MP₂ (isoquant curve)

Cost minimization implies producing a certain amount of output (y > 0) with the lowest possible cost. As with the maximization of profits, the firm is getting the most out of the resources it is using.

What is the difference between cost minimization and profit maximization? Cost minimization is a necessary condition for the maximization of profits, but not the other way around. For example, if you bring costs down to zero then output and profits will be zero too, failing to maximize profits while minimizing costs. As before, when... Continue reading "Cost Minimization and Profit Maximization in Economics" »

Pocket money isn’t a controversial topic nowadays.

If there have been a lot of debates over pocket money.

Pocket money used to be a controversial topic.

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Pocket money has always been a controversial topic.

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The debate regarding pocket money nowadays is about how much money should be given.

If today’s debate rages about something entirely different.

The debate regarding pocket money nowadays isn’t about how much money should be given.

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The debate regarding pocket money nowadays is about at what age money should be given.

If

Twelve-year-old will be able to use the new debit card.

A new debit card has just been launched for teenagers as young as 13.

Thirteen-year-olds won’t be able to use the credit card.

Teenagers over 13 will be able to use the... Continue reading "The Future of Pocket Money: Debit Cards and Financial Management" »

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

Revenue and Cost Formulas

• Revenue = Selling price per unit × Number of units sold
• Variable costs = Variable cost per unit × Number of units sold
• Total costs = Fixed costs + Variable costs
• Profit = Total revenue − Total costs or Total contribution − Fixed costs

Market and Investment Metrics

• Market capitalisation = Number of issued shares × Current share price
• Expected value (A & B) = (Pay-off of A × Probability of A) + (Pay-off of B × Probability of B)
• Net gain = Expected value − Initial cost of decision
• Market growth (%) = (Change in market size / Original market size) × 100
• Market share (%) = (Sales of one product or business / Total sales in the market) × 100
• Added value = Sales revenue − Cost of bought-in goods and services

Operational

CLARE FGF

1.- It was a long time since she had B carried

2.- Asked Bruno to address her C close

3.- A strong feeling of excitement D the average

4.- Having something to eat and B month compared

5.- Having something in common C growth, although

6.- She thought he might react B figure obtained

7.- Interested only in how she A researches put

8.- She discovers a way of D greater awareness

9.- Is in C line

RA:TS

10.- Nails A tend

9.- Record. D that

10.- Exception. H parti

11.- Sigh. F with

12.- Moment. A two

13.- To. C nowd HTTDFB

14.- Hugely. G it

15.- Strange. E when

14.- Country, Something

15.- Of such

WP

16.- Regularly B

17.- No longer C

18.- Tries to D

19.- Accepts C

20.- Is disa A

21.- Prefers D

22.- Is not A

23.- Now B

24.- Now D

25.- Likes to tell D

26.- Goes C -------