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,Convergenc
2、e,Convergencein probabilityprobability one or always sure,Convergence,Convergencein 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 Numbe
3、r (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)Marcinkie
4、wicz-Zygmund theorem,Common Statistical Distributions the Moments,The mean or ExpectationThe k-th momentVarianceThe standard deviationModeMedian -quantileP-interquantile rangleFunction of Random g(X),Common Statistical Distributions the Moments -,From Mathematica,Combinatorial Functions,the factoria
5、l functionthe binomial coefficientthe multinomial coefficientthe Catalan numbersthe Fibonacci numbersthe Fibonacci polynomials Fibonaccithe harmonic numbers the harmonic numbers of order r,Combinatorial Functions,the Bernoulli polynomialsthe Bernoullithe Euler polynomials the Euler numbersthe Genocc
6、hi numbersStirling numbers of 1rstStirling 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 sum of positive int
7、egers, 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 epsilon symbol).Clebsc
8、h-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 functionRiemann-Siege
9、l functionsStieltjes constants the 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, corresponding to value 0, occur
10、s 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 Statistical Distributions Hyper
11、-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-binomial - the distrib
12、ution 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,Poisson - describes the
13、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, the random variable X/(
14、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 variable tanX follows a Ca
15、uchy 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 sum of squares of v
16、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,Inver
17、ted-Chi-Squared,Common Statistical Distributions Squared-root Inverted-Gamma,Squared-root Inverted-Gamma,Common Statistical Distributions - Pareto,Pareto,Common Statistical Distributions Inverted-Pareto,Inverted-Pareto,Common Statistical Distributions - Normal,Normal / Gaussian,Common Statistical Di
18、stributions - Student,Student,Common Statistical Distributions Snedecor F,Snedecor F,Common Statistical Distributions - Logistic,Logistic - used in place of the normal distribution when a distribution with longer tails is desired.,Common Statistical Distributions - Weibull,Weibull - used in engineer
19、ing to describe the lifetime of an object (e.g., alpha=1.5 beta=3.5 in fig.).,Common Statistical Distributions - Laplace,Laplace - the distribution of the difference of two independent random variables with identical exponential distributions.,Common Statistical Distributions - Rayleigh,Rayleigh,Com
20、mon Statistical Distributions - extreme value distribution,The extreme value distribution, the limiting distribution for the largest values in large samples drawn from a variety of distributions, including the normal distribution; the limiting distribution for the smallest values in such samples can
21、 be obtained by multiplying Extreme Value Distribution random variables by -1. referred to as the log-Weibull distribution because of logarithmic relationships between an extreme value distributed random variable and a properly shifted and scaled Weibull distributed random variable,Common Statistica
22、l Distributions - halfnormal,Halfnormal - proportional to the distribution Normal Distribution with 0, 1/(theta Sqrt2/Pi) limited to the domain 0, ).,Common Statistical Distributions - Mutinomial,A k-variate multinomial distribution with index n and probability vector theta may be used to describe a
23、 series of n independent trials, in each of which just one of mutually exclusive events is observed with probability , thetak,Common Statistical Distributions - Negative Mutinomial,A k-variate negative multinomial distribution with success count n and failure probability vector theta may be used to
24、describe a series of n independent trials, in each of which there may be a success or one of mutually k exclusive modes of failure. The tth failure mode is observed with probability ,thetat , and the trials are discontinued when n successes are observed,Common Statistical Distributions - Poisson,A k
25、-variate multiple Poisson distribution with mean vector =0 +1, 0 +k is a common way to generalize the uni-variate Poisson distribution. Here the random k-vector X=X1,X k following this distribution is equivalent to Y=Y1,Y k, where Yt is a Poisson random variable with mean t, t=0,1,.,k.,Common Statis
26、tical Distributions - Dirichlet,Common Statistical Distributions - Multinomial-Dirichlet,Common Statistical Distributions - Multi-Normal,A k-variate multinormal distribution with mean vector mu and covariance matrix lambda is denoted N(mu,sigma2),Common Statistical Distributions - Multivariate Stude
27、nt,A vector that has a multivariate Student t distribution can also be written as a function of a multinormal random vector. Let X be a standardized multi-normal vector with covariance matrix R and let S2 be a chi-square variable with m degrees of freedom. Then X divided by S with sqrtm has a multiv
28、ariate distribution with correlation matrix R and m degrees of freedom, denoted t(R,m).,Common Statistical Distributions - Multivariate Wishart,If Xt is distributed multivariate normal with the zero mean and the covariance matrix Sigma, and X denotes the data matrix composed of the row vectors Xt, t
29、hen the matrix X has a Wishart distribution with scale matrix Sigma and m degrees of freedom parameter. The Wishart distribution is most typically used when describing the covariance matrix of multi-normal samples.,Common Statistical Distributions - Normal-Gamma,Common Statistical Distributions - Mu
30、ltivariate Normal-Gamma,Common Statistical Distributions - Multivariate Normal-Wishart,Common Statistical Distributions - Multi-Normal,A k-variate multinormal distribution with mean vector mu and covariance matrix lambda is denoted N(mu,sigma2),Common Statistical Distributions - Bilateral Pareto,End of Chapter 2,
