Probability - Index | Statistical Laboratory

1 ProbabilityAbout these people have written excellent notes for introductorycourses in Probability . Mine draw freely on material prepared by others in present-ing this course to students at Cambridge. I wish to acknowledge especially GeoffreyGrimmett, Frank Kelly and Doug order I follow is a bit different to that listed in the Schedules. Most of the materialcan be found in the recommended books by Grimmett & Welsh, and Ross. Many of theexamples are classics and mandatory in any sensible introductory course on book by Grinstead & Snell is easy reading and I know students have enjoyed are also some very good Wikipedia articles on many of the topics we will these notes I attempt a Goldilocks path by being neither too detailed or too brief.

2 Each lecture has a title and focuses upon just one or two ideas. My notes for each lecture are limited to 4 also include some entertaining, but nonexaminable topics, some of which are unusualfor a course at this level (such as random permutations, entropy, reflection principle,Benford and Zipf distributions, Erd os s probabilistic method, value at risk, eigenvaluesof random matrices, Kelly criterion, Chernoff bound).You should enjoy the book of Grimmett & Welsh, and the notes notes of notes, good or bad?I have wondered whether it is helpful or not topublish full course notes. On balance, I think that it is. It is helpful in that we candispense with some tedious copying-out, and you are guaranteed an accurate there are also benefits to hearing and writing down things yourself during a lecture,and so I recommend that you still do some of will say things in every lecture that are not in the notes.

3 I will sometimes tell you whenit would be good to make an extra note. In learning mathematics repeated exposureto ideas is essential. I hope that by doing all of reading, listening, writing and (mostimportantly) solving problems you will master and enjoy this recommend Tom K orner s treatise on how to listen to a maths these notesiTable of ContentsiiSchedulesviLearning outcomesvii1 Classical Diverse notions of Probability .. Classical Probability .. Sample space and events .. Equalizations in random walk ..32 Combinatorial Counting .. Sampling with or without replacement .. Sampling with or without regard to ordering .. Four cases of enumerative combinatorics ..83 Stirling s Multinomial coefficient.

4 Stirling s formula .. Improved Stirling s formula ..134 Axiomatic Axioms of Probability .. Boole s inequality .. Inclusion-exclusion formula ..175 Bonferroni s inequalities .. Independence of two events .. Independence of multiple events .. Important distributions .. Poisson approximation to the binomial ..216 Conditional Conditional Probability .. Properties of conditional Probability .. Law of total Probability .. Bayes formula .. Simpson s paradox ..24ii7 Discrete random Continuity ofP.. Discrete random variables .. Expectation .. Function of a random variable .. Properties of expectation ..298 Further functions of random Expectation of sum is sum of expectations.

5 Variance .. Indicator random variables .. Reproof of inclusion-exclusion formula .. Zipf s law ..339 Independent random Independent random variables .. Variance of a sum .. Efron s dice .. Cycle lengths in a random permutation ..3710 Jensen s inequality .. AM GM inequality .. Cauchy-Schwarz inequality .. Covariance and correlation .. Information entropy ..4111 Weak law of large Markov inequality .. Chebyshev inequality .. Weak law of large numbers .. Probabilistic proof of Weierstrass approximation theorem .. Benford s law ..4512 Probability generating Probability generating function .. Combinatorial applications ..4813 Conditional Conditional distribution and expectation.

6 Properties of conditional expectation .. Sums with a random number of terms .. Aggregate loss distribution and VaR .. Conditional entropy ..53iii14 Branching Branching processes .. Generating function of a branching process .. Probability of extinction ..5615 Random walk and gambler s Random walks .. Gambler s ruin .. Duration of the game .. Use of generating functions in random walk ..6116 Continuous random Continuous random variables .. Uniform distribution .. Exponential distribution .. Hazard rate .. Relationships among Probability distributions ..6517 Functions of a continuous random distribution of a function of a random variable .. Expectation .. Stochastic ordering of random variables.

7 Variance .. Inspection paradox ..6918 Jointly distributed random Jointly distributed random variables .. Independence of continuous random variables .. Geometric Probability .. Bertrand s paradox .. Buffon s needle ..7319 normal normal distribution .. Calculations with the normal distribution .. Mode, median and sample mean .. distribution of order statistics .. Stochastic bin packing ..7720 Transformations of random Transformation of random variables .. Convolution .. Cauchy distribution ..8121 Moment generating What happens if the mapping is not 1 1? .. Minimum of exponentials is exponential .. Moment generating functions .. Gamma distribution .. Beta distribution .

8 8522 multivariate normal Moment generating function of normal distribution .. Functions of normal random variables .. Bounds on tail Probability of a normal distribution .. multivariate normal distribution .. Bivariate normal .. multivariate moment generating function ..8923 Central limit Central limit theorem .. normal approximation to the binomial .. Estimating with Buffon s needle ..9324 Continuing studies in Large deviations .. Chernoff bound .. Random matrices .. Concluding remarks ..97A Problem solving strategies98B Fast Fourier transform and The Jacobian101D Beta distribution103E Kelly criterion104F Ballot theorem105G Allais paradox106H IB courses in applicable mathematics107 Index107 Richard Weber, Lent Term 2014vThis is reproduced from the Faculty this material will be covered in lectures, but in a slightly different concepts: Classical Probability , equally likely outcomes.

9 Combinatorial analysis, per-mutations and combinations. Stirling s formula (asymptotics for logn! proved). [3]Axiomatic approach: Axioms (countable case). Probability spaces. Inclusion-exclusionformula. Continuity and subadditivity of Probability measures. Independence. Binomial,Poisson and geometric distributions. Relation between Poisson and binomial Probability , Bayes formula. Examples, including Simpson s paradox. [5]Discrete random variables: Expectation. Functions of a random variable, indicator func-tion, variance, standard deviation. Covariance, independence of random variables. Generatingfunctions: sums of independent random variables, random sum formula, moments. Conditionalexpectation.

10 Random walks: gambler s ruin, recurrence relations. Difference equations andtheir solution. Mean time to absorption. Branching processes: generating functions and ex-tinction Probability . Combinatorial applications of generating functions. [7]Continuous random variables: Distributions and density functions. Expectations; expec-tation of a function of a random variable. Uniform, normal and exponential random property of exponential distribution . Joint distributions: transformation of ran-dom variables (including Jacobians), examples. Simulation: generating continuous randomvariables, independent normal random variables. Geometrical Probability : Bertrand s para-dox, Buffon s needle. Correlation coefficient, bivariate normal random variables.