Summary - Probability

Lecture 1: Course Introduction, Complex Numbers, and Functions of Complex Variables

Complex Numbers:

  • Definition: , where and are real numbers, and is the imaginary unit ().
  • Representation: Algebraic Form , Polar Form where and is the argument.
  • Operations: Addition, Multiplication, Conjugation, Modulus.

Functions of a Complex Variable:

  • Analytic Functions: Functions differentiable in a neighborhood.
  • Cauchy-Riemann Equations: Conditions for differentiability.
  • Line Integrals: Integrating a function along a curve.

Important Theorems:

  • Cauchy Theorem: Integral of an analytic function around a closed curve is zero.
  • Cauchy Integral Formula: Finding function values inside a curve.

Lecture 2: Analytic Functions and Cauchy-Riemann Equations

Analytic Functions:

  • Definition: Functions with a complex derivative at every point in some region.
  • Examples: Polynomials, Exponential functions, Trigonometric functions.
  • Uses: Solving complex integrals, conformal mappings, physics and engineering problems.

Lecture 3: Complex Integration

Line Integrals:

  • Definition: Integral of a complex function over a path.
  • Special Case: Path Independence - depends only on the endpoints, not the path.
  • Applications: Evaluating complex functions over curves, fluid dynamics, electromagnetism.

Lecture 4: Series Representation

Taylor Series:

  • Definition: Function approximation near a point.
  • Use:

Laurent Series:

  • Definition: Function representation in an annulus.
  • Special Case: Dealing with singularities.
  • Use:

Lecture 5: Residue Theorem and Applications

Residue Theorem:

  • Definition: Integral evaluation involving singularities.
  • Residue: Coefficient in Laurent series.
  • Applications: Evaluating complex integrals, calculating potential fields.

Lecture 6+7: Distributions and Fourier Transform

Distributions:

  • Definition: Generalized functions for integrating products.
  • Example: Dirac delta function .

Fourier Transform:

  • Definition: Converts a time function to a frequency function.
  • Use:
  • Applications: Signal processing, image analysis, solving partial differential equations.

Lecture 8: Laplace Transform

Laplace Transform:

  • Definition: Converts a time function to a complex function.
  • Use:
  • Applications: Solving differential equations, control theory, systems analysis.

Lecture 9: Probability Basics

Definitions:

  • Random Variable: Variable representing outcomes of a random process.
  • Probability Distribution: Describes the likelihood of different outcomes.
  • Example: Discrete (number of heads in coin tosses), Continuous (heights of people).

Lecture 10+11: Discrete and Continuous Probability Distributions

Discrete Distributions:

  • PMF: - probability of a discrete random variable being exactly a value.
  • Example: Binomial distribution.

Continuous Distributions:

  • PDF: - function where area under the curve represents the probability of a range.
  • Example: Normal distribution.

Lecture 12+13: Expectation and Variance

Expectation:

  • Definition: Average value of a random variable.
  • Use:
    (discrete),
    (continuous).

Variance:

  • Definition: Measure of how much values differ from the mean.
  • Use:
  • Applications: Fundamental in statistics and probability for describing distributions.

Lecture 14: Joint Distributions and Independence

Joint Distributions:

  • Definition: Probability of two random variables occurring together.
  • Example: Joint distribution of height and weight.

Independence:

  • Definition: Two variables and are independent if
  • Use: Simplifying the analysis of complex systems.

Probability

Lecture 1: Probability Fundamentals

Basic Concepts:

  • Experiment: Process of making an observation.
  • Outcome: Result of an experiment.
  • Sample Space (): Set of all possible outcomes.
  • Event: Subset of the sample space.

Probability:

  • Definition: Measure of the likelihood of an event.
  • Axioms: Non-negativity, Normalization, Additivity.

Lecture 2: Conditional Probability and Independence

Conditional Probability:

  • Definition: Probability of an event given another event has occurred.
  • Formula:

Independence:

  • Definition: Two events are independent if

Lecture 3: Discrete Random Variables

Random Variables:

  • Discrete Random Variable: Takes countable values.
  • PMF: Function that gives the probability of each outcome.
  • CDF: Function that gives the cumulative probability.

Expectation and Variance:

  • Expectation:
  • Variance:

Lecture 4: Continuous Random Variables

Random Variables:

  • Continuous Random Variable: Takes an uncountable range of values.
  • PDF: Function describing the probability density.
  • CDF: Cumulative distribution function.

Expectation and Variance:

  • Expectation:
  • Variance:

Lecture 5: Joint Distributions

Joint Probability:

  • Definition: Probability distribution over two random variables.
  • Joint PMF/PDF: Function giving the joint probability.

Covariance and Correlation:

  • Covariance: Measure of how two variables change together.
  • Correlation: Normalized measure of covariance.

Lecture 6: Limit Theorems

Law of Large Numbers:

  • Definition: Average of results from many trials will be close to the expected value.

Central Limit Theorem:

  • Definition: Sum of a large number of independent random variables will be approximately normally distributed.
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