Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Probability and Statistics


Probability and statistics are areas of mathematics that focus on the study of uncertainty and the analysis of data. They are essential to almost all fields of science, engineering, economics, and many other subjects. Understanding these concepts helps us make data-driven decisions and predictions. Here, we will explore these ideas in depth, using examples, visual aids, and simple explanations to clarify concepts.

Understanding probability

Probability is the branch of mathematics that deals with the likelihood or probability of a certain event occurring. It is important for predicting outcomes under uncertain circumstances.

Basic concepts of probability

Probability is measured on a scale from 0 to 1:

  • Probability 0 means the event will not occur.
  • Probability 1 means that the event will definitely happen.

The probability of an event is calculated using this formula:

Probability(Event) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

Example: tossing a coin

When you flip a fair coin, there are two possible outcomes: heads or tails. If you want to calculate the probability of getting heads, you would use the following formula:

Probability(Heads) = 1 / 2 = 0.5
Head Tail

Probability in throwing dice

Let's consider a six-sided die. Each face of the die represents an equally likely outcome. The probability of a specific number, such as 3, coming up is:

Probability(Rolling a 3) = 1 / 6 ≈ 0.1667
1 2 3 4 5 6

Advanced probability concepts

Conditional probability

Conditional probability is the probability of an event occurring, provided that another event has already occurred. The formula is:

P(A | B) = P(A and B) / P(B)

where P(A | B) is the probability of event A occurring, provided event B has already occurred.

Example: card draw

Imagine that you draw a card from a standard deck and know that it is a red card. The probability that it is a heart, provided that it is a red card, is:

P(Heart | Red) = P(Heart and Red) / P(Red) = 13/52 / 26/52 = 1/2

Bayes' theorem

Bayes' theorem is a fundamental concept of probability that describes the probability of an event based on prior knowledge of related events. It is expressed as:

P(A | B) = [P(B | A) * P(A)] / P(B)

Statistics: collecting and analyzing data

Descriptive statistics

Descriptive statistics summarize data. They provide simple summaries about the sample and measurement. Here are some key concepts:

Meaning

The mean, or average, is the sum of all data points divided by the number of points:

Mean = (Sum of all values) / (Number of values)

Median

The median is the middle value in a list ordered from smallest to largest. If the list has an even number of observations, the median is the average of the two middle numbers.

Method

The mode is the value that appears most often in a data set. A data set may have one mode, more than one mode, or no mode.

Standard deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values are close to the average, while a high standard deviation means that the values are spread over a wide range.

Inferential statistics

Inferential statistics are techniques that allow us to use samples to make generalizations about the populations from which the samples were taken.

Hypothesis testing

Hypothesis testing is a method that uses statistical evidence from sample data to estimate the truth of a hypothesis about a population parameter. It involves the main steps:

  1. Formulate the null and alternative hypotheses.
  2. Selecting a significance level.
  3. Calculating the test statistic and p-value.
  4. Deciding whether to accept or reject the null hypothesis.

Confidence interval

A confidence interval is a range of values that is likely to contain the population parameter of interest. A 95% confidence level implies that if the same population were sampled many times, the true parameter would lie within this interval 95% of the time.

Conclusion

Probability and statistics provide us with the tools necessary to analyze data and make predictions about various events. With a strong understanding of these concepts, you can solve problems systematically and arrive at more accurate conclusions.


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