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The chapter covers the Cumulative Distribution Function (CDF), outlining its significance in probability theory and its applications in various engineering fields, particularly when addressing uncertainties and probabilistic boundary conditions related to Partial Differential Equations (PDEs). It explains the definitions and properties of CDFs for both discrete and continuous random variables and highlights their relationship with Probability Density Functions (PDFs). Applications in heat transfer, reliability engineering, and stochastic PDEs emphasize the importance of CDFs in engineering analysis.
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Sections
References
unit 3 ch8.pdfClass Notes
Memorization
What we have learnt
- The Cumulative Distribution...
- CDFs differ in form for dis...
- CDFs are vital in modeling ...
Final Test
Revision Tests
What we have learnt
- The Cumulative Distribution Function defines the probability structure of a random variable.
- CDFs differ in form for discrete and continuous variables; discrete CDFs are step functions while continuous CDFs are integrals of their PDFs.
- CDFs are vital in modeling uncertainties in various engineering applications, linking stochastic processes with deterministic PDEs.
Key Concepts
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Term: Cumulative Distribution Function (CDF)
Definition: A function that indicates the probability that a random variable takes on a value less than or equal to a specific number.
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Term: Probability Density Function (PDF)
Definition: A function that represents the likelihood of a continuous random variable taking on a particular value, integral of which yields the CDF.
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Term: Discrete Random Variable
Definition: A random variable that can take on a countable number of distinct values, each with an associated probability.
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Term: Continuous Random Variable
Definition: A random variable that can take on any value within a given interval, described by a probability density function.
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Term: Stochastic Processes
Definition: Mathematical objects that evolve over time in a probabilistic manner, often modeled using CDFs and PDFs.