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

Probability Distributions

Discrete Random Variables

1. If the outcomes of a discrete random variable follow a Poisson distribution, then which of the following is true?
   • (A) The mean equals the variance.
2. Which of the following is not a characteristic of a binomial probability distribution?
   • (A) Each trial has a finite number of possible outcomes.
3. If the probability of a machine producing a defective part is 0.05, what is the probability of finding exactly 5 defective parts from a sample of 100? (Assume that the process follows a binomial distribution and round the answer to four places.)
   • (C) 0.1800
4. The number of arrivals of delivery trucks per hour at a loading station is an example of which of the following processes?
   • (C) Poisson
5. When sampling without replacement

Standard Form/Scientific Notation

If you had a number like this:

54300000.0

Then you would have to make the number so it is under 10:

5.4300000

But you moved the decimal place 7 places to the left, so the standard form is:

5.43 x 10⁷

HINT

Recurring decimals are written with a little dot or a line above a number to show that it goes on forever.

HINT

Terminating decimals are decimals that eventually stop and are not recurring.

Rounding

When rounding, you round to the nearest number with the amount of decimal places provided. For example:

59.54

The critical digit (last number) determines what the number will be. In this example, the critical digit is 4. 4 or less, let it rest. 5 or more, raise the score. So because the critical digit is 4, you would round the... Continue reading "Decimals, Fractions, and Percentages" »

Apothem

The perpendicular distance from the center of a regular polygon to a side of the polygon.

Center of a Circle

The point inside a circle that is the same distance from every point on the circle.

Center of a Regular Polygon

The point that is equidistant from all vertices of a regular polygon.

Central Angle of a Regular Polygon

An angle whose vertex is the center of the regular polygon and whose sides pass through consecutive vertices.

Circle

The set of points in a plane that are a fixed distance from a given point called the center of the circle.

Composite Figure

A plane figure made up of triangles, rectangles, trapezoids, circles, and other simple shapes, or a three-dimensional figure made up of prisms, cones, pyramids, cylinders, and other simple... Continue reading "Geometry Terms and Definitions" »

Financial Concepts and Calculation Practice

Expectation Theory and Two-Year Rates

Problem: If the one-year rate on an instrument is 9.0% and the expected rate for a one-year instrument one year from today is 9.89%, what is the expected two-year rate today, if the expectation theory holds?

• One-year rate: 9.00%
• One-year rate, one year forward: 9.89%

Formula: (1 + R1) * (1 + ER2) = (1 + R2)^2

Calculation:

• (1 + 0.09) * (1 + 0.0989) = (1 + R2)^2
• (1.09) * (1.0989) = (1 + R2)^2
• 1.197801 = (1 + R2)^2
• sqrt(1.197801) = 1 + R2
• 1.094449 = 1 + R2
• R2 = 0.094449

Expected Two-Year Rate: 9.44% (approximately 9.4%)

Understanding the Rule of 72

The Rule of 72 is a quick method to estimate the number of years it takes for an investment to double in value, given a fixed annual... Continue reading "Essential Financial Concepts & Calculations" »

Which of the following is most likely an issuer bonds: HEDGE FUND

A bond issued by a city would most likely be classified as a: NON-SOVEREIGN GOVERNMENT BOND

A fixed-income security issued with a maturity at issuance of nine months is most likely classified: MMSECURITY

The price of a bond issued in the US by a British company and denominated in US dollars is most likely: CHANGE AS US INTEREST RATES CHANGE

Interbank offered rates are best described as the rates at which major banks can: BORROW UNSECURED FUNDS

A company issues floating-rates bonds. The coupon rates is expressed as the three-month Libor plus a spread. The coupon payments are most likely: LIBOR INCREASES

A 10-year bond was issued four years ago. The bond is denominated in US dollars,

Taylor Series

f(x+h) = f(x) + f'(x)h + f''(x)h² /2! + f'''(x)h³ /3! + ...

Secant Method (Find Root)

1. Rewrite as f(x)=0
2. Set initial points
3. f'(x₂) = (f(x₂) - f(x₁)) / (x₂-x₁)
4. New point becomes x₂
5. Iterate

LU Decomposition

A = LU

Crout's

Identity top

Doolittle

Identity bottom

1. Set up LU
2. Find components through matrix multiplication and solve for variables

Solving for b matrix

LUX=B

1. Set UX = Y
2. Solve LY = B using algebra
3. Solve UX = Y using algebra

Norm

Absolute value of the largest row sum

Condition

||A||_inf||A^-1||_inf

Gauss-Seidel

1. Set each equation to a variable
2. Use any initial values for the first equation
3. Use the x₁ from the first equation to solve x₂
4. Iterate (Fast version of Jacobi Method)

Discrete Least Squares Approximation

AX = B

1. Make matrix A by [ n ∑(x) ∑(x²) ...]
2. Make

E-Business and Its Impact

The Changing Landscape of Business

The rise of the internet has dramatically transformed the business landscape, giving rise to the concept of e-business. E-business encompasses a wide range of activities, including B2B (business-to-business) and B2C (business-to-consumer) transactions, supply chain management, and customer relationship management.

Key Characteristics of E-Business

• Integration: E-business systems integrate various business functions and channels, such as brick-and-mortar stores, online platforms, and mobile apps.
• Flexibility: E-business frameworks are designed to be flexible, open, and adaptable to changing market conditions.
• Ubiquity: E-business infrastructure enables ubiquitous commerce, allowing businesses

Single-Factor ANOVA is used to determine whether three or more populations have equal means.

ANOVA Assumptions:

• The population values are normally distributed.
• The variances for each population are equal.
• The samples are independent.
• The data measurement is interval or ratio level.

ANOVA is an analysis of variance. It needs to have equal variance in order to test the means.

Single-Factor ANOVA: analysis of variance design in which independent samples are obtained from two or more levels of a single factor for the purpose of testing whether the levels have equal means.

Type of data used in ANOVA: ratio or interval data (numbers).

H₀: all means are equal.

H₁: there is a difference in means.

Theory: Practicality.

In Multifactor ANOVA:

• Groups: rows (router

IFRS for SMEs: Accounting Policies, Estimates, and Errors

Understanding Accounting Policies, Estimates, and Prior Period Errors

Statement Evaluation:

• Statement: "While accounting for particular transactions, events, or conditions, entities are allowed to deviate from basic requirements imposed by IFRS for SMEs, but only if the effect of following the existing IFRS for SMEs guidance will not be material."
• Truthfulness: This statement is False. IFRS for SMEs generally requires adherence to its principles. Deviation is not permitted simply based on materiality; materiality primarily affects disclosure, not the application of recognition or measurement principles.

What Changes are Treated as Changes in Accounting Policy?

A change is treated as a change... Continue reading "IFRS for SMEs: Accounting Policies, Estimates, and PPE Principles" »

Fixed Costs

Fixed costs are not a function of the output. They do not vary with the output. They cannot be avoided until the operation of the firm is closed. They are contractual (prime).

Recurring and Non-Recurring Costs

• Recurring: Predetermined expenses for running the business, e.g., salaries, repairs, maintenance.
• Non-Recurring: Not predetermined, regular budgeting, e.g., repair of a machine. Sometimes it is also planned.

Cost Estimation

Cost estimation is the process of finding an estimate, an approximation of a value, which will be used for some purpose, though it is completely uncertain and unstable. Estimation is typically a value from statistics used to estimate the value of a corresponding population parameter.

Learning Curve

The learning... Continue reading "Cost Estimation and Management Decision Areas in Business" »