Table of Contents

1. Introduction
2. What Is Group Theory?
3. Definitions: Groups, Subgroups, and Cosets
4. Homomorphisms and Isomorphisms
5. Group Actions and Orbits
6. Symmetry in Physics
7. Permutation Groups and Parity
8. Lie Groups and Lie Algebras
9. Representations of Groups
10. Irreducible Representations and Characters
11. SU(2) and Spin
12. SU(3) and the Quark Model
13. SO(3), Rotations, and Angular Momentum
14. Noether’s Theorem and Conservation Laws
15. Applications in Quantum Mechanics, Field Theory, and Crystallography
16. Conclusion

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1. Introduction

Group theory is the study of symmetry. In physics, it provides the language for describing conservation laws, particle classifications, quantum numbers, and more. From fundamental interactions to solid-state crystals, group theory offers a unifying structure behind physical phenomena.

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2. What Is Group Theory?

A group is a set \( G \) with an operation \( * \) that satisfies:

1. Closure: \( \forall a,b \in G, \ a * b \in G \)
2. Associativity: \( a * (b * c) = (a * b) * c \)
3. Identity: \( \exists e \in G, \ e * a = a * e = a \)
4. Inverses: \( \forall a \in G, \exists a^{-1} \in G, \ a * a^{-1} = e \)

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3. Definitions: Groups, Subgroups, and Cosets

• Subgroup: a subset of a group that is itself a group
• Cosets: partitions of a group based on a subgroup
• Lagrange’s Theorem: order of a subgroup divides order of the group

Examples:

• \( \mathbb{Z}_n \): integers modulo \( n \)
• \( S_n \): permutation group on \( n \) elements

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4. Homomorphisms and Isomorphisms

• Homomorphism: structure-preserving map between groups
  \[
  \phi(ab) = \phi(a)\phi(b)
  \]
• Isomorphism: bijective homomorphism; shows structural equivalence

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5. Group Actions and Orbits

A group action is a rule for applying elements of a group to objects in a set.

• Orbits: set of all objects reachable by the group action
• Stabilizers: elements of the group that fix an object

Used to classify symmetric states and degenerate configurations.

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6. Symmetry in Physics

Symmetries correspond to invariance under transformations:

• Translational symmetry → conservation of momentum
• Rotational symmetry → conservation of angular momentum
• Gauge symmetry → conservation of charge

Symmetry breaking explains phase transitions and mass generation.

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7. Permutation Groups and Parity

• Permutation groups \( S_n \) model discrete symmetries
• Parity transformation: spatial inversion \( \vec{x} \to -\vec{x} \)
  Important in weak interactions and particle classification

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8. Lie Groups and Lie Algebras

Lie groups are continuous groups (e.g., rotations):

• Smooth manifolds + group structure
• Examples: \( U(1), SU(2), SU(3), SO(3), SO(1,3) \)

Lie algebra: tangent space at identity with commutators:

\[
[T_a, T_b] = i f_{abc} T_c
\]

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9. Representations of Groups

A representation is a map from group elements to matrices:

\[
g \mapsto D(g), \quad D(g_1 g_2) = D(g_1)D(g_2)
\]

Used to study group action on vector spaces (e.g., quantum states).

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10. Irreducible Representations and Characters

• A representation is irreducible if it has no invariant subspaces
• Character: trace of representation matrix:

\[
\chi(g) = \text{Tr}(D(g))
\]

Character tables help classify particle states and identify symmetry sectors.

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11. SU(2) and Spin

• \( SU(2) \): group of 2×2 unitary matrices with determinant 1
• Double cover of \( SO(3) \), related to spin and angular momentum
• Spin-1/2 particles (e.g., electrons) described by SU(2) representations

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12. SU(3) and the Quark Model

• \( SU(3) \) describes internal symmetry of quarks (color or flavor)
• Eight generators → eight gluons (in QCD)
• Used in the Eightfold Way and Gell-Mann’s classification

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13. SO(3), Rotations, and Angular Momentum

• SO(3): special orthogonal group of 3D rotations
• Quantum angular momentum algebra:

\[
[J_i, J_j] = i \hbar \epsilon_{ijk} J_k
\]

Eigenstates classified by \( j \) and \( m \) quantum numbers

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14. Noether’s Theorem and Conservation Laws

Noether’s theorem:

Every continuous symmetry of a physical system corresponds to a conserved quantity.

Examples:

• Time translation → energy conservation
• Space rotation → angular momentum
• Gauge symmetry → electric charge

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15. Applications in Quantum Mechanics, Field Theory, and Crystallography

• Quantum mechanics: symmetry groups classify spectra and operators
• Quantum field theory: gauge groups define interactions (e.g., \( SU(3) \times SU(2) \times U(1) \))
• Crystallography: space groups determine allowed lattice structures

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16. Conclusion

Group theory provides a powerful framework to understand the fundamental symmetries of nature. From discrete permutations to continuous Lie groups, it connects the abstract structure of mathematics to the tangible laws of physics.

Its language is essential for modern theoretical physics, from particle models to condensed matter systems.

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Table of Contents

1. Introduction
2. What Is a Stochastic Process?
3. Index Sets and State Spaces
4. Types of Stochastic Processes
5. Stationarity and Ergodicity
6. Markov Processes
7. Discrete-Time Markov Chains
8. Continuous-Time Markov Processes
9. Poisson Processes
10. Birth-Death Processes
11. Brownian Motion and Wiener Process
12. Stochastic Differential Equations (SDEs)
13. Fokker–Planck and Langevin Equations
14. Martingales and Filtration
15. Applications in Physics, Finance, and Biology
16. Conclusion

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1. Introduction

A stochastic process is a mathematical model describing systems that evolve over time with an inherent randomness. Unlike deterministic systems, the future state of a stochastic process cannot be predicted exactly, only in terms of probability distributions.

Stochastic processes appear across physics, biology, finance, and engineering — from quantum measurements to stock prices and population dynamics.

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2. What Is a Stochastic Process?

Formally, a stochastic process is a collection of random variables indexed by time (or space):

\[
\{X(t) : t \in T\}
\]

Where:

• \( T \) is the index set (e.g., time)
• \( X(t) \) is a random variable describing the system state at time \( t \)

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3. Index Sets and State Spaces

• Index set: can be discrete (e.g., \( t = 0, 1, 2, \dots \)) or continuous (e.g., \( t \in [0, \infty) \))
• State space: the set of all possible values of \( X(t) \), which can be finite, countable, or continuous

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4. Types of Stochastic Processes

• Discrete-time vs. Continuous-time
• Discrete-state vs. Continuous-state
• Markovian vs. non-Markovian
• Stationary vs. non-stationary

Classification helps choose appropriate models and solution techniques.

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5. Stationarity and Ergodicity

• Stationary process: statistical properties (mean, variance) are invariant over time
• Ergodic process: time averages equal ensemble averages

Stationarity simplifies analysis, especially in signal processing and statistical mechanics.

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6. Markov Processes

A process has the Markov property if the future depends only on the present, not on the past:

\[
P(X_{t+1} | X_t, X_{t-1}, \dots) = P(X_{t+1} | X_t)
\]

This memoryless property enables elegant mathematical treatment.

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7. Discrete-Time Markov Chains

Described by a transition probability matrix \( P \):

\[
P_{ij} = P(X_{n+1} = j \mid X_n = i)
\]

Analysis focuses on:

• Transition probabilities
• Stationary distributions
• Absorbing states
• Recurrence and transience

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8. Continuous-Time Markov Processes

Described by infinitesimal generators or rate matrices \( Q \):

\[
\frac{d}{dt} P(t) = QP(t)
\]

Applications include:

• Chemical reactions
• Queueing systems
• Epidemic models

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9. Poisson Processes

A fundamental counting process:

• \( N(t) \): number of events by time \( t \)
• Inter-arrival times are exponential with rate \( \lambda \)
• Independent, memoryless increments

Used to model:

• Arrivals in queues
• Radioactive decay
• Network traffic

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10. Birth-Death Processes

A class of continuous-time Markov processes where transitions occur between neighboring states:

\[
P_{n,n+1} = \lambda_n, \quad P_{n,n-1} = \mu_n
\]

Models:

• Population growth
• Queue lengths
• Chemical kinetics

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11. Brownian Motion and Wiener Process

A continuous-time, continuous-state stochastic process:

• Starts at 0
• Has independent, normally distributed increments
• Continuous paths, nowhere differentiable

Mathematical model of diffusion:

\[
B(t) \sim \mathcal{N}(0, t)
\]

Foundation for stochastic calculus.

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12. Stochastic Differential Equations (SDEs)

SDEs describe dynamics of systems influenced by noise:

\[
dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dB_t
\]

Where \( dB_t \) is Brownian motion (Wiener process) noise.

Applications:

• Finance (Black–Scholes model)
• Physics (Brownian motion, Langevin dynamics)

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13. Fokker–Planck and Langevin Equations

• Langevin equation: stochastic differential equation for velocity/momentum
• Fokker–Planck equation: governs evolution of probability density:

\[
\frac{\partial P(x,t)}{\partial t} = -\frac{\partial}{\partial x}[A(x)P] + \frac{1}{2} \frac{\partial^2}{\partial x^2}[B(x)P]
\]

Describes time evolution of stochastic systems in terms of densities.

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14. Martingales and Filtration

A martingale is a process where the conditional expected future equals the present:

\[
\mathbb{E}[X_{t+1} \mid \mathcal{F}_t] = X_t
\]

Important in finance and gambling theory. Filtration represents growing information over time.

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15. Applications in Physics, Finance, and Biology

• Physics: diffusion, statistical mechanics, quantum noise
• Finance: option pricing, risk modeling, interest rate models
• Biology: gene expression noise, population dynamics, neural activity
• Engineering: signal processing, queueing theory

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16. Conclusion

Stochastic processes provide a rich mathematical framework to study dynamic systems influenced by chance. From Brownian motion to stock prices and genetic drift, they underpin modern scientific modeling across disciplines.

A deep understanding of these processes is essential for research in applied mathematics, theoretical physics, finance, and systems biology.

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Table of Contents

1. Introduction
2. What Is a Statistical Distribution?
3. Types of Distributions: Discrete vs Continuous
4. Probability Mass Functions (PMFs)
5. Probability Density Functions (PDFs)
6. Cumulative Distribution Function (CDF)
7. Bernoulli and Binomial Distributions
8. Poisson Distribution
9. Geometric and Negative Binomial Distributions
10. Uniform Distribution (Discrete and Continuous)
11. Normal (Gaussian) Distribution
12. Exponential and Gamma Distributions
13. Beta Distribution
14. Chi-Square and t-Distributions
15. Multivariate Distributions
16. Applications in Science and Engineering
17. Conclusion

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1. Introduction

Statistical distributions are mathematical functions that describe the likelihood of different outcomes in a random experiment. They are foundational in probability theory, statistical inference, machine learning, and physical sciences.

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2. What Is a Statistical Distribution?

A distribution assigns probabilities or densities to all possible outcomes of a random variable. It allows us to:

• Quantify uncertainty
• Model data behavior
• Infer population characteristics

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3. Types of Distributions: Discrete vs Continuous

• Discrete distributions: finite or countable outcomes (e.g., dice rolls)
• Continuous distributions: uncountably infinite outcomes (e.g., time, height)

Each has its own mathematical description and properties.

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4. Probability Mass Functions (PMFs)

Used for discrete random variables:

\[
P(X = x_i) = p_i, \quad \sum_i p_i = 1
\]

Examples: Bernoulli, Binomial, Poisson

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5. Probability Density Functions (PDFs)

Used for continuous random variables:

\[
P(a \le X \le b) = \int_a^b f(x) dx, \quad \int_{-\infty}^\infty f(x) dx = 1
\]

Examples: Normal, Exponential, Gamma

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6. Cumulative Distribution Function (CDF)

The CDF gives the probability that a variable takes a value less than or equal to \( x \):

\[
F(x) = P(X \le x)
\]

For continuous variables:
\[
F(x) = \int_{-\infty}^x f(t) dt
\]

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7. Bernoulli and Binomial Distributions

• Bernoulli: \( X \in \{0, 1\} \), success probability \( p \)
• Binomial: \( n \) Bernoulli trials, success probability \( p \)

PMF of binomial:

\[
P(X = k) = \binom{n}{k} p^k (1 – p)^{n-k}
\]

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8. Poisson Distribution

Models rare events over time/space:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

Where \( \lambda \) is the average rate.

Used in:

• Queueing
• Radioactive decay
• Web traffic modeling

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9. Geometric and Negative Binomial Distributions

• Geometric: trials until first success
• Negative Binomial: trials until \( r \)-th success

Geometric PMF:

\[
P(X = k) = (1 – p)^{k-1} p
\]

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10. Uniform Distribution (Discrete and Continuous)

• Discrete: equal probability for each outcome
• Continuous: PDF is constant over interval:

\[
f(x) = \frac{1}{b – a}, \quad a \le x \le b
\]

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11. Normal (Gaussian) Distribution

Central in probability theory:

\[
f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{(x – \mu)^2}{2\sigma^2} \right)
\]

Characterized by:

• \( \mu \): mean
• \( \sigma^2 \): variance

Arises naturally via the central limit theorem.

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12. Exponential and Gamma Distributions

• Exponential: time between events:

\[
f(x) = \lambda e^{-\lambda x}, \quad x \ge 0
\]

• Gamma: generalization with shape \( k \):

\[
f(x) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}
\]

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13. Beta Distribution

Used for modeling probabilities (bounded on [0,1]):

\[
f(x) = \frac{x^{\alpha – 1}(1 – x)^{\beta – 1}}{B(\alpha, \beta)}
\]

Where \( B \) is the beta function. Very flexible shape.

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14. Chi-Square and t-Distributions

• Chi-square: sum of squared standard normals
  Used in hypothesis testing and confidence intervals
• t-distribution: accounts for small sample sizes; converges to normal as \( n \to \infty \)

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15. Multivariate Distributions

Describes multiple random variables:

• Joint distribution: \( f(x, y) \)
• Multivariate normal: generalization of Gaussian to vector-valued random variables

Applications in:

• Machine learning
• Statistical physics
• Bayesian networks

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16. Applications in Science and Engineering

• Physics: Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein
• Signal processing: noise modeling
• Biostatistics: survival analysis
• Finance: stock returns and risk models
• Machine learning: generative models, Naive Bayes

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17. Conclusion

Statistical distributions are the mathematical backbone of modeling uncertainty. Mastering their properties and applications allows scientists and engineers to analyze data, simulate systems, and infer hidden patterns in the natural world.

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