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Chapter 1: Probability and Measure

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Masters Stochastic Processes and FInance Flashcards on Chapter 1: Probability and Measure, created by mho1 on 06/06/2015.
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Flashcards by mho1, updated more than 1 year ago
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Created by mho1 almost 10 years ago
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Question Answer
\sigma-algebra
Image: 87a8e6a7-044d-45cc-a891-b72d3528a1e8 (image/png)
probability measure
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measurable function
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random variable/random vector
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probability law/probability distribution
Image: d42bd6f2-afe4-486a-bf4c-c98c19af2741 (image/png)
cumulative distribution function
Image: 6dc201ef-78a8-4c2e-a2c3-60141be801fa (image/png)
probability density function
Image: 1cb413a6-cf60-4b44-b287-1c8ff9e56252 (image/png)
random variables X and Y are identically distributed if...
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sub-\sigma-algebra
Image: cd2d77b1-38de-4532-aa51-896122d01052 (image/png)
independent (\sigma-algebra)
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independence (random variables)
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Lebesgue integration Step 1: Indicator functions
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Lebesgue Integration Step 2: simple functions
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Lebesgue Integration Step 3: non-negative random variables
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Lebesgue Integration Step 4: arbitrary random variables
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integrable
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null set
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Proof - see lecture notes
Properties of the integral (five) linearity domination triangle inequality set bounded integrability change of variable
Properties of the Integral/Expectation 1. Linearity
Image: 11bbed44-4d81-4c40-a99f-da88656c38ac (image/png)
Properties of the Integral/Expectation 2. Domination
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Properties of the Integral/Expectation 3. the triangle inequality
Image: 604e64db-5f93-4f49-a88a-fdea52d43bb8 (image/png)
Properties of the Integral/Expectation 4. Set bounded integrability
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Properties of the Integral/Expectation 5. Change of variable
Image: 56da118a-a508-4a8f-87f7-0ca0050d8054 (image/png)
Dominated Convergence Theorem
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Monotone Convergence Theorem
Image: 22abb476-073f-4736-af2d-e93578588a3a (image/png)
nth moment mean variance
Image: 6e53d448-9208-4fc6-a49e-577ea5fe2933 (image/png)
moment generating function
Image: 3f951801-63da-4b8d-86f1-8db96e132a94 (image/png)
Almost sure convergence
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mean square convergence
Image: 446df0d6-4d8f-4554-93d8-b3954267a9fe (image/png)
Chebychev's Inequality
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Cauchy-Schwarz inequality
Image: a7c808c5-2464-45bd-90c7-2f4c2d2caebc (image/png)
Image: 7cd9d29c-abc5-42d5-9ca7-16ab2e234fa1 (image/png)
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convex function
Image: 437b1e72-f196-4be5-a334-3c4573d75af4 (image/png)
Jensen's inequality
Image: ab4e4d37-feb8-4d97-a115-9679f32cd777 (image/png)
Image: 103a4d89-a3f8-442c-942d-18d82a6f7908 (image/png)
Proof - see notes
Absolutely Continuous measure Equivalent measures
Image: dd1a54e5-5f1f-4fe1-9125-48e32873ec8e (image/png)
Radon-Nikodym
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Radon-Nikodym derivative
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stochastic process
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integrable stochastic process
Image: 22fc68be-f1fd-4f5c-857a-044f08c8f1cd (image/png)
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