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 Theory ↓


Probability Distributions


Probability distributions are fundamental concepts in the fields of probability theory and statistics. They describe how probabilities are distributed over the values of a random variable. Probability distributions can be used to model real-world phenomena, represent statistical data, and they form the backbone for statistical inference.

Understanding random variables

A random variable is a variable whose possible values are the numerical outcomes of a random event. There are two main types of random variables: discrete and continuous.

  • Discrete random variable: A random variable with a countable number of possible values. For example, the number of students in a class or the number of times a coin comes up heads when tossed.
  • Continuous random variable: A random variable that can take on infinite possible values. For example, the height of students in a class or the time taken to run a marathon.

Types of probability distributions

  1. Discrete probability distributions
  2. Continuous probability distributions

Discrete probability distributions

Discrete probability distributions are used to deal with discrete random variables. One of the most common discrete probability distributions is the binomial distribution.

Binomial distribution

The binomial distribution models the number of successes in a certain number of independent Bernoulli trials, each of which has the same probability of success. For example, consider tossing a coin ten times. The probability of getting a particular number of heads is represented by the binomial distribution.

The probability mass function (PMF) of the binomial distribution is given as follows:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

  • C(n, k) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials
  • p is the probability of success in a single trial
  • k is the number of successes
  • n is the total number of trials
0 1 2 Binomial PMF

Continuous probability distributions

Continuous probability distributions apply to continuous random variables. A well-known continuous probability distribution is the normal distribution.

Normal distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its symmetric bell shape. It is defined by its mean (µ) and standard deviation (σ).

The probability density function (PDF) of the normal distribution is:

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

This function describes how the probability distribution is distributed across different values of x.

Normal PDF

Properties of probability distributions

There are several important properties of probability distributions:

  • Sum of probabilities: For a discrete distribution, the sum of all probabilities in the PMF must equal 1. For a continuous distribution, the integral of the PDF over the entire space must equal 1.
  • Mean (expected value): Represents the average or expected outcome. For discrete distributions: E(X) = ∑ x * P(x); For continuous distributions: E(X) = ∫ x * f(x) dx.
  • Variance: Measures the spread of outcomes of a random variable. For discrete distributions: Var(X) = ∑ (x - µ)² * P(x); For continuous distributions: Var(X) = ∫ (x - µ)² * f(x) dx.

Applications of probability distributions

Probability distributions are important in many fields such as finance, engineering, science, and others. For example, in finance, the normal distribution is often used in modeling stock prices or returns. In manufacturing, the binomial distribution can be used to model defects in batches of products.

Another common example is using the Poisson distribution for modeling the number of events occurring in a given time interval, such as the number of cars crossing a bridge per hour or the decay events occurring per unit time from a radioactive source.

The versatility and ubiquity of these distributions make them essential tools for analysts, scientists, and statisticians who want to make informed predictions and understand the underlying mechanisms of various phenomena.

Conclusion

Understanding probability distributions is crucial for anyone delving deeper into the fields of statistics and probability. From their foundational role in undergraduate-level mathematics to their practical applications in real-world problems, mastering probability distributions provides a diverse toolbox for modeling uncertainty and making predictions.


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Probability Distributions

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