This section introduces the fundamental concepts of probability, random variables, and stochastic processes. Probability theory is the mathematical foundation for understanding uncertainty and randomness. Random variables are functions that assign numerical values to outcomes of random experiments. Stochastic processes are collections of random variables indexed by time or space.
Key Concepts
- Probability Space: A probability space is a mathematical construct that consists of a sample space, a σ-algebra, and a probability measure. It provides a framework for assigning probabilities to events.
- Random Variable: A random variable is a function that maps outcomes of a random experiment to real numbers. There are two types of random variables: discrete and continuous.
- Probability Distribution: The probability distribution of a random variable describes the probabilities of different values that the variable can take.
- Stochastic Process: A stochastic process is a collection of random variables indexed by time or space. It models systems that evolve over time or space.
Applications
Probability theory and stochastic processes have wide-ranging applications in various fields, including:
- Engineering: Reliability analysis, queuing theory, and risk assessment.
- Finance: Option pricing, portfolio optimization, and risk management.
- Statistics: Hypothesis testing, confidence intervals, and regression analysis.
- Computer Science: Machine learning, artificial intelligence, and data analysis.
Related Resources
For further reading and resources on probability, random variables, and stochastic processes, please visit the following links:
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