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 ↓


Random Variables


Introduction

In probability theory and statistics, random variables serve as a fundamental concept. They represent quantities that can take different values, each of which has an associated probability. Understanding random variables is important in the study of probability, as they help us model and analyze random events in a structured way.

Defining random variables

A random variable is essentially a function that assigns a numerical value to each outcome in a sample space. A sample space is the set of all possible outcomes of a random experiment. For example, if you roll a six-sided dice, the sample space is {1, 2, 3, 4, 5, 6}.

Types of random variables

Random variables are generally classified into two main types:

  • Discrete random variables: These variables take on a countable number of distinct values. An example of this is the outcome of throwing a dice, which can only result in one of a limited number of possibilities: 1, 2, 3, 4, 5, or 6.
  • Continuous random variables: These variables take on infinite possible values. For example, the exact amount of rain in a day can be any non-negative real number.

Visual example: discrete random variable

Consider a simple random experiment of tossing a coin twice. The sample space is:

{HH, HT, TH, TT}

Define the random variable X as the number of heads that appear. Again, X can take the values 0, 1, or 2.

The probability distribution of X can be represented visually as follows:

0 1 2 0.25 0.5 0.25

Visual example: continuous random variable

For a continuous random variable, consider an example where the random variable Y represents the height of a randomly chosen tree in a forest. This random variable can take any value within a given range.

The probability distribution of Y will often be represented as a smooth curve:

The area under the curve between any two points represents the probability of the random variable falling within that range.

Probability mass function (PMF) and probability density function (PDF)

Random variables are characterized by functions that describe the probabilities associated with their possible values. For discrete random variables, this is represented by the probability mass function (PMF). For continuous random variables, we use the probability density function (PDF).

Probability mass function (PMF)

The PMF of a discrete random variable X is a function p(x) that gives the probability P(X = x) for each value x in the sample space.

Probability density function (PDF)

The PDF of a continuous random variable Y is a function f(y) such that the probability that Y falls within an interval [a, b] is given by the integral of f(y) over that interval:

P(a ≤ Y ≤ b) = ∫ a b f(y) dy

Properties of random variables

Random variables have properties that help summarize and understand their behavior. These include the expected value, variance, and standard deviation.

Expected value

The expected value or mean of a random variable is a measure of its central tendency. For a discrete random variable X with PMF p(x), the expected value E(X) is:

e(x) = Σ x * p(x)

For a continuous random variable Y with pdf f(y), the expected value E(Y) is:

e(y) = ∫ y * f(y) dy

Variance and standard deviation

The variance provides a measure of how spread out the values of a random variable are around the mean. The variance of X is given by:

Var(X) = E((X - E(X))^2)

The standard deviation is the square root of the variance and gives a measure of spread in the same units as the random variable.

Example: discrete random variable calculation

Suppose we have a random variable Y that represents the outcomes of rolling a fair six-sided dice. The values of Y are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6.

The expected value of Y is:

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

Example: continuous random variable calculation

Consider a random variable Z that follows a uniform distribution between 0 and 3. The PDF is defined as:

f(z) = 1/3, for 0 ≤ z ≤ 3

The expected value of Z is:

e(z) = ∫ 0 3 z * (1/3) dz

Solving this integral gives E(Z) = 1.5.

Conclusion

Random variables are an important component of probability and statistics, which provide a way to measure and analyze random phenomena. Understanding the characteristics and properties of random variables equips us with the tools necessary to explore a wide range of real-world problems where uncertainty and randomness are present.


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Random Variables

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