第三届丘成桐大学生数学竞赛试题-概率统计及应用数学.pdf

INDIVIDUAL TEST S T YAU COLLEGE MATH CONTESTS 2012 Probability and Statistics Please solve 5 out of the following 6 problems or highest scores of 5 problems will be counted 1 Solve the following two problems 1 An urn contains b black balls and r red balls One of the balls was drawn at random and putted back in the urn with a additional balls of the same color Now suppose that the second ball drawn at random is red What is the probability that the fi rst ball drawn was black 2 Let Xn be a sequence of random variables satisfying lim a sup n 1 P Xn a 0 Assume that sequence of random variables Yn converges to 0 in prob ability Prove that XnYn converges to 0 in probability 2 Solve the following two problems 1 Let F P be a probability space G be a sub algebra of F Assume that X is a non negative integrable random variable Set Y E X G Prove that a X 0 Y 0 a s b Y 0 ess inf A A G X 0 A 2 Let X and Y have a bivariate normal distribution with zero means variances 2and 2 respectively and correlation Find the condi tional expectation E X X Y 3 Suppose that p i j i 1 2 m j 1 2 n is a fi nite bivariate joint probability distribution that is p i j 0 m X i 1 n X j 1 p i j 1 i Can p i j be always expressed as p i j X k kak i bk j for some fi nite k 0 P k k 1 ak i 0 Pm i 1ak i 1 bk j 0 Pn j 1bk j 1 1 2 ii Prove or disprove the above relation by use of conditional prob ability 4 Let X1 Xmbe an independent and identically distributed i i d random sample from a cumulative distribution function CDF F and Y1 Ynan i i d random sample from a CDF G We want to test H0 F G versus H1 F 6 G The total sample size is N m n Consider the following two nonparametric tests The Wilcoxon rank sum tests The test proceeds by fi rst rank ing the pooled X and Y samples and then taking the sum of the ranks associated with the Y sample Let Ry1 Rynbe the rankings of the sample y1 ynfrom the pooled sample in increasing order The Wilcoxon rank sum statistic is defi ned as W Pn j 1Ryj The Mann Whitney U test Let Uij 1 if Xi Yj and Uij 0 otherwise The Mann Whitney U statistic is defi ned as U Pm i 1 Pn j 1Uij The probability P X Y can be estimated as U mn The decision rule is based on assessing if 0 5 Assume that there are no tied data values a Show that W U 1 2n n 1 which shows that the two test statistics diff er only by a constant and yield exactly the same p values b Using the central limit theorem the Wilcoxon rank sum statis tic W can be converted to a Z variable which provides an easy to use approximation The transformation is ZW W W W where Wand 2 W are the mean and variance of W under H0 Show that W 1 2n N 1 and 2 W 1 12mn N 1 5 Let X be a random variable with EX2 and Y X Assume that X has a Lebesgue density symmetric about 0 Show that random variables X and Y are uncorrelated but they are not independent 6 Let E1 Enbe i i d random variables with Ei Exponential 1 Let U1 Unbe i i d uniformly on 0 1 distributed random vari ables Further assume that E1 Enand U1 Unare indepen dent a Find the density of X E1 Em E1 En where m m diff erent points xkand f x is smooth a prove that there is a unique solution Px x to 2 b denote h maxk xk x prove ci 1 i f i x C f i hm 1 i i 0 1 m where f i is the i th derivative of f and C f i denote some constant depending on f i c if S xk k 1 2 K are symmetrically distributed around x that is if xk S then 2x xk S prove that ci 1 i f i x C f i hm 2 i i 0 1 m 1 2 for i 0 1 m with the same parity of m 3 Describe the forward in time and center in space fi nite diff erence scheme for the one wave wave equation ut ux 0 i Conduct the von Neumann stability analysis and comment on their stability property ii Under what condition on t and x would this scheme be stable and convergent iii How many ways you can modify this scheme to make it stable when the CFL condition is satisfi ed 4 Let C and D in Cn nbe Hermitian matrices Denote their eigen values by 1 2 nand 1 2 n respectively Then it is known that n X i 1 i i 2 kC Dk2 F 1 Let A and B be in Cn n Denote their singular values by 1 2 nand 1 2 n respectively Prove that the following inequality holds n X i 1 i i 2 kA Bk2 F 2 Given A Rn nand its SVD is A U V T where U u1 u2 un V v1 v2 vn are orthogonal matrices and diag 1 2 n 1 2 n 0 Suppose rank A k and denote by Uk u1 u2 uk Vk v1 v2 vk k diag 1 2 k and Ak Uk kV T k k X i 1 iuivT i Prove that min rank B k kA Bk2 F kA Akk2 F n X i k 1 2 i 3 3 Let the vectors xi Rn i 1 2 n be in the space W with dimension d where d n Let the orthonormal basis of W be W Rn d Then we can represent xiby xi c Wri ei i 1 2 n where c Rnis a constant vector ri Rdis the coordinate of the point xiin the space W and eiis the error Denote R r1 r2 rn and E e1 e2 en Find W R and c such that the error kEkFis minimized Hint write X x1 x2 xn c 1 1 1 WR E 5 Two primes p and q are called twin primes if q p 2 For example 5 and 7 1