Notes, summaries, assignments, exams, and exercises for Mathematics for University

The Expectation-Maximization (EM) algorithm is an iterative method used to estimate the parameters of statistical models that involve latent (unobserved) variables, such as missing data or hidden cluster assignments. It is especially useful for fitting Gaussian Mixture Models (GMMs), where the goal is to model data as a mixture of several Gaussian distributions.

How the EM Algorithm Works

The EM algorithm alternates between two steps:

• Expectation Step (E-step): Given the current parameter estimates (means, covariances, and mixing coefficients for GMMs), the algorithm computes the probability (or "responsibility") that each data point belongs to each Gaussian component. This step essentially fills in the missing information about which cluster

Simple Random Sample

A sample designed in such a way as to ensure that (1) every member of a population has an equal chance of being included in the sample, and (2) every sample of size n has the same chance of occurring.

Selecting Elements

Subjects are selected at fixed sampling intervals by selecting every kth element thereafter. The interval between selections is determined by the ratio of the population size to the sample size (N/n). For example, if the population size is N=1,000 and a sample size of n=100 is desired, then the sampling interval is 1,000/100 = 10, so every 10th person is selected into the sample.

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

Understanding Prefixes and Suffixes

A prefix is a letter or a group of letters that appears at the beginning of a word and changes the word’s original meaning.

A suffix is a letter or a group of letters that is usually added onto the end of words to change the way a word fits into a sentence grammatically.

Nouns, verbs, adjectives, and adverbs all tend to use different suffixes, which makes them easier to remember. Collectively, prefixes and suffixes are known as affixes.

What Are Affixes?

An affix is added to the root of a word to change its meaning.

• An affix added to the front of a word is known as a prefix.
• One added to the back is known as a suffix.

Sometimes, prefixes are hyphenated. Read more about hyphens in prefixes.

Examples of Affixes

Here...

Monthly Financial Data

10 reasons to use games in language teaching

1.Games create a context for meaningful communication. 2. This meaningful communication serves as a basis for comprehensible input. 3. Games add interest to what learners find boring. 4. Games can be used with all language skills. 5. Games offer a fun experience. 6. Games encourage participation from all learners. 7. Games are learner-centered activities. 8. Games work outside of class. 9. Games promote cooperative learning. 10. Games fit into multiple intelligence theory.

Using Games in Teaching English to Young Learners: How to Choose a Game

Students may wish to play games purely for fun. Teachers need more convincing reasons. The key to a successful language game is that the rules are clear. The...

This analysis delves into the financial performance of the company for the years 2007 and 2008, focusing on key liquidity and operational efficiency metrics such as working capital and various maturity periods.

Working Capital Calculation (2007-2008)

The working capital is calculated as the difference between current assets and current liabilities. Here are the results for 2007 and 2008:

• 2008: Current Assets (2,788,286) - Current Liabilities (3,020,313) = -232,027
• 2007: Current Assets (2,741,769) - Current Liabilities (2,953,480) = -211,711

Both periods show negative working capital. This indicates that the company's current liabilities exceed its current assets. The negative working capital remained almost constant across both years.

Analysis of

Secondary Data Analysis: Sources and Elaboration

Secondary data analysis considers both qualitative and quantitative perspectives. It uses references like coding sheets in quantitative designs and identifies interviews or focus groups for qualitative data. Official statistics and public sources are used to verify field data.

Desrosieres argues that statistical indicators from official sources reflect social reality in institutions. These indicators are presented without analysis, as tables of results. This means the original data has been elaborated, which is typical of secondary sources. Secondary sources have developed, summarized, and synthesized the data. The degree of elaboration is the key difference between secondary data analysis and...

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...

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:...