# Chapter 1: Probability and Measure

Flashcards by mho1, updated more than 1 year ago
 Created by mho1 over 5 years ago
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### Description

Masters Stochastic Processes and FInance Flashcards on Chapter 1: Probability and Measure, created by mho1 on 06/06/2015.

## Resource summary

 Question Answer \sigma-algebra probability measure measurable function random variable/random vector probability law/probability distribution cumulative distribution function probability density function random variables X and Y are identically distributed if... sub-\sigma-algebra independent (\sigma-algebra) independence (random variables) Lebesgue integration Step 1: Indicator functions Lebesgue Integration Step 2: simple functions Lebesgue Integration Step 3: non-negative random variables Lebesgue Integration Step 4: arbitrary random variables integrable null set 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 Properties of the Integral/Expectation 2. Domination Properties of the Integral/Expectation 3. the triangle inequality Properties of the Integral/Expectation 4. Set bounded integrability Properties of the Integral/Expectation 5. Change of variable Dominated Convergence Theorem Monotone Convergence Theorem nth moment mean variance moment generating function Almost sure convergence mean square convergence Chebychev's Inequality Cauchy-Schwarz inequality convex function Jensen's inequality Proof - see notes Absolutely Continuous measure Equivalent measures Radon-Nikodym Radon-Nikodym derivative stochastic process integrable stochastic process

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