# Chapter 1.1 : Foundations – Mathematics and Statistics: (Basic Mathematics, Algebra and Calculus, Statistics, Probability, and Hypothesis testing)

Vidya Rajasekaran
@Vidya-Rajasekaran

22nd December 2023 | 6 Views

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### 1. Algebra:

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. Here are some fundamental algebraic concepts:

Variables and Expressions: Variables represent unknown or changing quantities. Expressions are combinations of variables, numbers, and operations (addition, subtraction, multiplication, division).

a + b (addition of two numbers)
a − b (subtraction of two numbers)
a × b or a * b (multiplication of two numbers)
a / b   (division of two numbers)

Example :

Equations and Inequalities: Equations express the equality of two expressions. Inequalities express relationships where one side is greater than or less than the other.

(linear equation)

(inequality)

Example :

Functions:  Functions describe relationships between variables. They have inputs (independent variable) and outputs (dependent variable).

(linear function)

Example :

### 2. Calculus:

Calculus involves the study of rates of change and the accumulation of quantities. Key concepts include:

Limits and Continuity: Limits describe the behavior of a function as the input approaches a certain value. Continuity ensures that there are no abrupt jumps or holes in the graph of a function.

(limit of a function)

is continuous at

Derivatives: Derivatives measure the rate at which a function changes. They represent slopes of tangent lines to curves.

or (derivative of )

(power rule)

Example :

Integrals: Integrals compute the accumulation of quantities and the area under curves. They are the reverse process of derivatives.

(indefinite integral)

Example :

### 3. Statistics:

Statistics involves the collection, analysis, interpretation, presentation, and organization of data. Key concepts include:

Descriptive Statistics: Measures of central tendency (mean, median, mode). Measures of dispersion (range, variance, standard deviation).

Example :

Inferential Statistics: Uses sample data to make inferences about a population. Includes hypothesis testing and confidence intervals.

Example :

### 4. Probability:

Probability deals with the likelihood of events occurring. Key concepts include:

Probability Basics: Probability is a number between 0 and 1, representing the likelihood of an event. The sum of all probabilities in a sample space is 1.

Example :

Random Variables and Probability Distributions: Random variables represent outcomes of a random process. Probability distributions describe the likelihood of different outcomes.

Example :

### Hypothesis Testing:

Hypothesis testing is a statistical method to make inferences about a population based on a sample of data. Key steps include:

Formulating Hypotheses: Null hypothesis (H0) and alternative hypothesis (H1).

Example :

Choosing a Significance Level: Common levels include 0.05 or 0.01.

Example :

Collecting Data and Calculating Test Statistic: Use statistical tests (t-tests, chi-square tests) to analyze data.

Example :

Making a Decision: Compare the p-value to the significance level and make a decision about the null hypothesis.

Compare the p-value to the significance level (α)

Example :

Compare the p-value to ( to accept or reject the null hypothesis.

Vidya Rajasekaran

@Vidya-Rajasekaran

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