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 Theory ↓


Probability Distributions


Probability is a fascinating and practical area of mathematics that allows us to understand and quantify uncertainty. A fundamental concept within probability and statistics is the "probability distribution", which describes how probabilities are distributed over the possible outcomes of an experiment.

What is a probability distribution?

A probability distribution is a mathematical function that assigns the probabilities of occurrence of different possible outcomes in an experiment. Imagine you are tossing a fair coin. There are two possible outcomes: heads and tails. The probability of each outcome is 0.5. This simple scenario can be described by a probability distribution.

More formally, probability distributions can be defined for both discrete and continuous random variables. The outcomes of a discrete random variable are countable, while the number of outcomes of a continuous random variable is infinite. Let's learn about these in more detail.

Discrete probability distributions

Discrete probability distributions are relevant to a system that has a finite or countable number of possible outcomes. Each outcome has a probability associated with it, and the sum of all probabilities is 1. For example, tossing a coin is a discrete random variable with outcomes S = {heads, tails}, each of which has a probability:

P(Heads) = 0.5 P(Tails) = 0.5 Sum of probabilities = P(Heads) + P(Tails) = 1

Another typical example of a discrete probability distribution is the roll of a six-sided dice. The possible outcomes are numbers from 1 to 6, and a fair dice gives the probability of each face:

P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Sum of probabilities = P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1
123456

Continuous probability distributions

In contrast, a continuous random variable has infinite possible values. Think of measuring the height of a randomly selected person. The height can be any value within a certain range. For continuous distributions, probabilities are assigned to intervals rather than to specific outcomes.

The most common continuous probability distribution is the normal distribution, often referred to as the "bell curve". It is characterized by two parameters: the mean (μ) and the standard deviation (σ). The probability density function (PDF) for the normal distribution is:

f(x) = (1 / (σ √(2π))) * e^(-0.5 * ((x - μ) / σ)^2)
Mean (μ)

In this equation, 'e' represents the exponential function, and π is pi, which is approximately 3.14159. The normal distribution is symmetric around the mean, and probabilities are calculated as areas under the curve.

Properties of probability distributions

Whether we're dealing with discrete or continuous distributions, there are some fundamental properties that all probability distributions have in common:

  • Non-negativity: The probability of each outcome is always between 0 and 1.
  • Total Probability: The sum of the probabilities of all possible outcomes is always 1.
  • Generalization: In the case of continuous distribution, the total area under the probability density function is 1.

Expected value

The expected value (or mean) of a random variable provides a measure of the "center" of the distribution. It can be thought of as the average result if an experiment were repeated an infinite number of times. The expected value of a discrete random variable, X, is calculated as follows:

E(X) = Σ [x * P(x)]

For example, if X denotes the outcome of throwing a fair six-sided die, then:

E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5

The expected value for continuous variables uses a similar concept integrated over all possible outcomes:

E(X) = ∫ x * f(x) dx

Variance and standard deviation

The variance of a random variable provides a measure of the spread of the distribution. It tells us how far the results are from the expected value. The variance for a discrete random variable is calculated as follows:

Var(X) = Σ [(x - E(X))^2 * P(x)]

The standard deviation is simply the square root of the variance, which provides a measurement in the same units as the random variable.

Normal probability distribution

Binomial distribution

The binomial distribution is a discrete probability distribution. It is used to model the number of successful outcomes in a certain number of trials, where each trial has two possible outcomes (success or failure) and is independent of other trials. The probability mass function for the binomial distribution is given as:

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

Where 'n' is the number of attempts, 'k' is the number of successes, and 'p' is the probability of success.

Equal distribution

The uniform distribution is the simplest type of distribution. In the discrete case, each outcome is equally likely. The continuous uniform distribution describes an experiment where each outcome is equally likely within a specified range.

Poisson distribution

The Poisson distribution shows the number of times an event occurs in a given interval of time or space and is characterized by the parameter λ (lambda), which is the average number of events occurring in the interval.

P(X = k) = (λ^k * e^(-λ)) / k!

Applications of probability distributions

Probability distributions are widely used in a variety of fields, including finance, engineering, science, and everyday life. They help make decisions under uncertainty by quantifying risks, modeling real-world phenomena, and assisting in statistical inference.

For example, insurance companies use probability distributions to assess risk and set premiums. Engineers use them to model the reliability and failure rates of components. Economists rely on distributions to model consumer behavior and market dynamics.

Conclusion

Probability distributions provide a powerful tool for understanding the uncertainty and variability inherent in various processes and experiments. Whether working with discrete or continuous random variables, the concepts of expected value, variance, and normal distribution types are essential for mastering both probability theory and its applications.

By understanding these concepts, you will be able to not only perform calculations, but also gain insight into the events and phenomena described by probability models. Probability distributions allow us to make informed decisions in the face of uncertainty, which is an important aspect of both everyday life and professional fields.


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