Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate ↓


Probability and Statistics


Probability and statistics are branches of mathematics that deal with the concept of uncertainty. They are used to analyze and predict outcomes and make decisions based on data. While probability provides the theoretical framework for measuring uncertainty, statistics uses that framework to collect, analyze, interpret, and present empirical data.

Understanding probability

Probability is a measure of the likelihood of an event occurring. It measures our expectations about that event based on certain conditions or experiments. The probability of any event is a number between 0 and 1, where 0 represents impossibility and 1 represents certainty. Events with high probability are more likely to occur than events with low probability.

Consider a simple example. If you have a fair six-sided dice, the probability of getting a particular number, say 3, is:

Probability of rolling a 3 = 1/6 = 0.1667

This is because the dice has six sides and each side has an equal chance of falling face up.

Visual example: Probability of a single dice roll

1 2 3 4 5 6

Basic concepts of probability

Randomized experiment

A random experiment is a procedure whose outcome cannot be predicted with certainty in advance. For example, tossing a coin, throwing dice, or drawing a card from a shuffled deck. Despite their randomness, these experiments can become predictable after a long series of trials.

Sample locations and events

The sample space of a random experiment, often denoted by S, is the set of all possible outcomes. Each possible outcome is called a sample point. An event is any subset of the sample space. For example, in throwing a dice, the sample space S is {1, 2, 3, 4, 5, 6} and an event could be "rolling an even number", which includes the outcomes {2, 4, 6}.

Combination of events

Two or more events in the sample space can be combined using set operations such as union, intersection, and complement. For example, if A and B are two events, then:

  • A ∪ B (Union): Event occurs if either A or B or both occur.
  • A ∩ B (Intersection): The event occurs if both A and B occur.
  • A' (complement): The event occurs if A does not occur.

Laws of probability

Probability rules are rules that govern how probabilities can be assigned to events in a probability space. These include the axioms of probability, which are as follows:

  1. Non-negativity: The probability of any event A is greater than or equal to 0. 
    P(A) ≥ 0
  2. Generalization: The entire sample space S has probability 1. 
    P(S) = 1
  3. Additivity: For mutually exclusive events A and B, the probability of A or B occurring is the sum of their individual probabilities. 
    P(A ∪ B) = P(A) + P(B)

For more complex scenarios, conditional probability and Bayes' theorem are essential concepts:

Conditional probability

The probability of event A given that event B has occurred is called the conditional probability of A B denoted by P(A|B). It is calculated as follows: 

P(A|B) = P(A ∩ B) / P(B)
provided that P(B) > 0.

Bayes' theorem

Bayes' theorem deals with the conditional and marginal probabilities of random events. It is an important tool for updating probabilities based on new information. Bayes' theorem is expressed as: 

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

Random variables and probability distributions

Random variables

A random variable is a variable that takes on different numerical values depending on the outcome of a random experiment. Random variables are classified into discrete and continuous types.

Discrete random variables: These take a countable number of possible outcomes. For example, throwing a dice and counting the number of successes in a series of experiments.

Continuous random variables: These can take an infinite number of possible outcomes within a given range. For example, the exact height of persons or the time taken to complete a task.

Probability distributions

The probability distribution describes how probabilities are distributed over the values of a random variable. For a discrete random variable, this is known as the probability mass function (PMF): 

P(X = x) = p(x)
For a continuous random variable, this is known as the probability density function (PDF): 
f(x)
The area under the PDF curve for a given interval provides the probability that the random variable falls within that interval.

Visual example: Probability mass function

0.25 0.20 0.25 0.05

Normal probability distribution

Binomial distribution

The binomial distribution is a discrete distribution that describes the number of successes in a certain number of independent Bernoulli trials, with the same probability of success. The probability of exactly k successes in n trials is given by: 

P(X = k) = C(n, k) * p^k * (1-p)^(nk)
where C(n, k) is the binomial coefficient.

Normal distribution

The normal distribution, also known as the Gaussian distribution, is a continuous distribution that is symmetric about the mean. It is defined by its mean (µ) and standard deviation (σ) and is given by the probability density function: 

f(x) = (1/(σ√(2π))) * e^(-(x-µ)²/(2σ²))

Visual example: Normal distribution

Mean

Introduction to statistics

Statistics is the discipline that deals with the collection, organization, analysis, interpretation, and presentation of data. It has two main branches: descriptive statistics and inferential statistics.

Descriptive statistics

Descriptive statistics summarize and describe the main characteristics of a dataset. This includes measures of central tendency, dispersion, and graphical representation.

Measures of central tendency: These measures describe the center of a data set. Common measures include the mean, median, and mode.

Measures of dispersion: Dispersion represents the spread of data points. Common measures include range, variance, and standard deviation.

Inferential statistics

Inferential statistics uses random samples of data taken from a population to describe and make inferences about the population. It includes hypothesis testing, estimation, and prediction.

Hypothesis testing

Hypothesis testing is a method of making decisions using data obtained from a scientific study. It involves testing an assumption or claim about a population parameter.

For example, you may want to test whether a new drug is more effective than an existing drug. You form two hypotheses:

  • H0 (null hypothesis): There is no difference in effectiveness.
  • H1 (alternative hypothesis): The new drug is more effective.

Conclusion

Probability and statistics form the basis of data analysis and decision making in various fields including science, engineering, economics and others. Probability helps measure uncertainty and estimate the likelihood of outcomes in random experiments, while statistics helps explain real-world imperatives based on data collection and analysis. Understanding these concepts is essential for making informed decisions in an uncertain world.


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