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1、Quantitative Risk Management - Probability and Statistics,Ming-Heng Zhang,Purpose,Essentials of Probability of StatisticsImportant Distributions,Underlying,Assumption : X, X1, X2, is a sequence of iid non-degeneration rvs defined on a probability space , F, P with common DF.Cumulative sums,Convergen
2、ce,Convergence in probabilityprobability one or always sure,Convergence,Convergence in mean squarein distribution,Example -,ConvergenceRelationship,Inequality,Chebyshevs InequalityMakovs InequalityErgodic theorem/遍历性,Weak Law of Large Numbers - WLLN,Weak Law of Large Number (WLLN) or Law of Large Nu
3、mber (LLN)Criterion for the WLLN,Example -,Let X be symmetric with tail for some constant c0Check conditionConflict,The Strong Law of Large Numbers - SLLN,Strong Law of Large Number (SLLN)Criterion for the SLLN,The Central Limit Theorem - Application,Glivenko-Cantelli theorem(the empirical DF)Marcin
4、kiewicz-Zygmund theorem,Common Statistical Distributions the Moments,The mean or Expectation The k-th moment Variance The standard deviation Mode Median -quantile P-interquantile rangleFunction of Random g(X),Common Statistical Distributions the Moments -,From Mathematica,Combinatorial Functions,the
5、 factorial function the binomial coefficient the multinomial coefficient the Catalan numbers the Fibonacci numbers the Fibonacci polynomials Fibonacci the harmonic numbers the harmonic numbers of order r,Combinatorial Functions,the Bernoulli polynomials the Bernoullithe Euler polynomials the Euler n
6、umbers the Genocchi numbersStirling numbers of 1rst Stirling numbers of 2nd,Combinatorial Functions,The partition function - PartitionsPn gives the number of ways of writing the integer n as a sum of positive integers, without regard to order. PartitionsQn gives the number of ways of writing n as a
7、sum of positive integers, with the constraint that all the integers in each sum are distinct. The signature function gives the signature of a permutation. It is equal to +1 for even permutations (composed of an even number of transpositions), and to -1 for odd permutations (Levi-Civita symbol or eps
8、ilon symbol). Clebsch-Gordan coefficients 3-j symbols or Wigner coefficients 6-j symbols or Racah coefficients,Special Functions - Gamma and Related Functions,Gammas Functions,ContourPlotAbsGammax + i y, x, -3, 3, y, -2, 2, PlotPoints-50 ,Special Functions - Zeta and Related Functions,Riemann zeta f
9、unction Riemann-Siegel functions Stieltjes constantsthe generalized Riemann zeta function or Hurwitz zeta function,Common Statistical Distributions Bernoulli,the probability distribution for a single trial in which success, corresponding to value 1, occurs with probability p, and failure, correspond
10、ing to value 0, occurs with probability 1-p.,Common Statistical Distributions - Binomial,the distribution of the number of successes that occur in n independent trials, where the probability of success in each trial is p.,Common Statistical Distributions Binomial-Beta,Binomial-Beta,Common Statistica
11、l Distributions Hyper-geometric,Hyper-geometric - used in place of the binomial distribution for experiments in which the n trials correspond to sampling without replacement from a population of size ntotal with nsucc potential successes.,Common Statistical Distributions Negative-binomail,Negative-b
12、inomial - the distribution of the number of failures that occur in a sequence of trials before n successes have occurred, where the probability of success in each trial is p.,Common Statistical Distributions Negative-binomail-Beta,Negative-binomial-Beta,Common Statistical Distributions - Poisson,Poi
13、sson - describes the number of points in a unit interval, where points are distributed with uniform density m.,Common Statistical Distributions Poisson-Gamma,Poisson-Gamma,Common Statistical Distributions - Beta,Beta - When X and Y have independent gamma distributions with equal scale parameters, th
14、e random variable X/(X+Y) follows the beta distribution with parameters and , where and are the shape parameters of the gamma variables.,Common Statistical Distributions - Uniform,Uniform,Common Statistical Distributions - Cauchy,Cauchy - If X is uniformly distributed on -p, p, then the random varia
15、ble tanX follows a Cauchy distribution with a=0 and b=1.,Common Statistical Distributions - Gamma,Gamma,Common Statistical Distributions - Exponential,Exponential,Common Statistical Distributions Gamma-Gamma,Gamma-Gamma,Common Statistical Distributions Chi-Squared,Chi-Squared - the distribution of a
16、 sum of squares of v unit normal random variables. Also called as a chi-square distribution with v degrees of freedom.,Common Statistical Distributions Noncentral Chi-Squared,Non-central Chi-Squared,Common Statistical Distributions Inverted-Gamma,Inverted-Gamma,Common Statistical Distributions Inverted-Chi-Squared,Inverted-Chi-Squared,Common Statistical Distributions Squared-root Inverted-Gamma,Squared-root Inverted-Gamma,
