Chapter 1: Probability and Measure

Question	Answer
\sigma-algebra	Image: 87a8e6a7-044d-45cc-a891-b72d3528a1e8 (image/png)
probability measure	Image: fa9785ce-4c31-449a-8729-305ecbd8550a (image/png)
measurable function	Image: f84398bc-2650-4ab3-b79d-747eb6e9e3f9 (image/png)
random variable/random vector	Image: 09d67719-7d83-486d-8f65-780187c1b10a (image/png)
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...	Image: 8a128bd0-d93b-46fe-a8df-f1a834c9538f (image/png)
sub-\sigma-algebra	Image: cd2d77b1-38de-4532-aa51-896122d01052 (image/png)
independent (\sigma-algebra)	Image: 8ea098cb-ea2e-4df8-b650-0999b0bd1ff1 (image/png)
independence (random variables)	Image: 7c0049ba-c2a8-4ef4-a61c-2361325956f3 (image/png)
Lebesgue integration Step 1: Indicator functions	Image: e8fdadf8-1817-442c-809b-7c3b7383ffa6 (image/png)
Lebesgue Integration Step 2: simple functions	Image: 3018fdd8-2674-4c6c-93b1-3db798d3780e (image/png)
Lebesgue Integration Step 3: non-negative random variables	Image: 9981ca58-e58e-4f28-91d9-06200272bade (image/png)
Lebesgue Integration Step 4: arbitrary random variables	Image: faae9407-851b-4049-93e7-f4c5e88dc80d (image/png)
integrable	Image: 2bf6a67c-9b1f-49a5-82c5-b1deada0a992 (image/png)
null set	Image: b97a2855-7983-42d3-babb-2d609cfc0ead (image/png)
Image: 47b953a1-484e-4c43-8a75-baa76a8b5a9a (image/png)	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	Image: da4f9adf-4d3c-4c21-a50a-239be089ae45 (image/png)
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	Image: b02c7d37-94a4-48f0-91c4-4a253fe3d9cd (image/png)
Properties of the Integral/Expectation 5. Change of variable	Image: 56da118a-a508-4a8f-87f7-0ca0050d8054 (image/png)
Dominated Convergence Theorem	Image: 7caa6ea1-9b86-4c9b-a205-6358e6ae36a4 (image/png)
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	Image: eff3e683-a812-4aeb-94b6-a34a2884515f (image/png)
mean square convergence	Image: 446df0d6-4d8f-4554-93d8-b3954267a9fe (image/png)
Chebychev's Inequality	Image: 0c8b8296-0c76-447c-a22d-fe42cb480ce6 (image/png)
Image: 79191497-f7f5-4812-a688-402eda3b1021 (image/png)	Image: b2319ae9-18b2-42c8-b394-21a9597681b8 (image/png)
Cauchy-Schwarz inequality	Image: a7c808c5-2464-45bd-90c7-2f4c2d2caebc (image/png)
Image: 7cd9d29c-abc5-42d5-9ca7-16ab2e234fa1 (image/png)	Image: 99fd340a-98de-499c-9854-d6792e88531f (image/png)
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	Image: bcb0034b-fceb-4b8f-98d4-3bc324913311 (image/png)
Radon-Nikodym derivative	Image: cff23a9b-8647-45bf-88f6-eaede15f6fd4 (image/png)
stochastic process	Image: f99c22f0-f166-4182-b401-570e3e24e5ff (image/png)
integrable stochastic process	Image: 22fc68be-f1fd-4f5c-857a-044f08c8f1cd (image/png)

\sigma-algebra

Image: 87a8e6a7-044d-45cc-a891-b72d3528a1e8 (image/png)

probability measure

Image: fa9785ce-4c31-449a-8729-305ecbd8550a (image/png)

measurable function

Image: f84398bc-2650-4ab3-b79d-747eb6e9e3f9 (image/png)

random variable/random vector

Image: 09d67719-7d83-486d-8f65-780187c1b10a (image/png)

probability law/probability distribution

Image: d42bd6f2-afe4-486a-bf4c-c98c19af2741 (image/png)

cumulative distribution function

Image: 6dc201ef-78a8-4c2e-a2c3-60141be801fa (image/png)