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" »

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

Portfolio Theory and Expected Returns

1. Expected return of the portfolio: μ_P = w₁μ₁ + w₂μ₂.

Standard deviation of the portfolio return: σ_p = √(w₁²σ₁² + w₂²σ₂² + 2ρw₁w₂σ₁σ₂).

• Suppose two investments R₁ and R₂ with expected returns μ₁ and μ₂.
• w₁ is the proportion of money in the first investment.
• σ₁ and σ₂ are the respective standard deviations.
• ρ is the coefficient of correlation between the two investments.

Market Portfolio and Systematic Risk

2. Relationship between any investment and the market portfolio: R = a + βR_M + ε.

• R = return from investment.
• R_M = return from market portfolio.
• a and β are constants.
• ε is a random variable representing the regression error.
• βR_M: Systematic risk.
• ε: Non-systematic risk.

Capital Asset Pricing

What are Standard Data Systems (SDS)?

A Standard Data System (SDS) is a database of normal time values organized by work elements. It is used to establish time standards for tasks composed of those work elements. A key feature is that time standards can be established before the job is actually performed.

When is an SDS Suitable?

A Standard Data System is particularly suitable in situations involving:

• A high degree of similarity between tasks.
• Batch production environments.
• A large number of standards that need to be set.
• The need to set standards before production begins.

Using a Standard Data System

The process for using an SDS typically involves the following steps:

1. Analyze the new tasks and divide them into their constituent work elements.
2. Access

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