## Chapter 1: Preparatory material

### A review of probability theory

- measure space (with total measure one)
- events or random variables
- sample space a probability space : set + -algebra (closed under complement & countable unions and intersections ) of related subset + probability measure
- extension of the sample space
**probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space****union bound**: , also called**zeroth moment method**- asymptotic notations: large and small O
- measuring the probability of an event:
- surely: equal to the event
- almost surely: occurs with proba 1
- with overwhelming probability: for evert fixed , holds with proba
- with high proba: holds with proba for some independent of
- asymptotically almost surely: holds with proba , i.e., the pobab goes to 1 as .

**uniformly**in__some parameter__if the constant of bound is independent of the parameter.- random variables
- the affect of events of proba zero, as long as there are countable many of them
- joint random variable
- vector and matrix-valued r.v.
- point processes: e.g., the spectrum of eigenvalues of a random (square) matrix.
- distribution : probability measure/ law
- can we always create a r.v. with a specific distribution? extension may be necessary
- discrete distribution examples:
- Dirac distribution
- discrete uniform distribution
- (unsigned) Bernoulli distribution
- signed Bernoulli distribution
- Geometric distribution
- Possion distribution

- probability density function (PDF) : Radon-Nikodym theorem tells us that there exists non-negative and absolutely integrable such that
- cumulative distribution function (CDF) : indeed is the Lebesgue-Stieltjes measure of
- examples of absolutely continuous scalar distribution:
- uniform distribution
- real normal distribution: standard normal distribution
- complex normal distribution

- expectation of mean (positive case) and by Fubini-Tonelli theorem , of course can be extended to R or C.
- expectation is linear and monotone if unsigned
**Markov inequality**, also referred to as**first moment method**: but often gives suboptimal results since it does not explore any independence in the system**Borel-Cantelli lemma**: if then a.s. at most finitely many of the events occurs at once.- expectation of function of a r.v.
- moments
- exponential moments
- Fourier moments/ characteristic function
- resolvent
- negative moment
- zeroth moment

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