1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơ...
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance drawn (independently) from a Gaussian distribution with mean μ and convariance Σ. Recall /IML Xm, and Show that EML]-NN Σ Y ou mav want to prove, then use . where àn,m = 1 if m n and = 0 otherwise.
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance...
14&15
13 Let o be a permutation of a set A. We shall say "o moves a € A" if o(a) are moved by a cycle o E SA of length n? a. If A is a finite set, how many elements 14 Let A be an infinite set. Let H be the set of all o ESA such that the number of elements moved by o is finite. Show that H is a subgroup of SA
Question 6: Let n 2 1 be an integer and let A[1...n] be an array that stores a permutation of the set { 1, 2, . .. , n). If the array A s sorted. then Ak] = k for k = 1.2. .., n and, thus. TL k-1 If the array A is not sorted and Ak-i, where iメk, then Ak-서 is equal to the "distance" that the valuei must move in order to make the array sorted. Thus,...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of the set (1, 2, . . . ,n). Define ao = 0 and an+1 = 0, and consider the sequence do, 1, d2, l3, . . . , Un, Un+1 A position i with 1 i n is called auesome, if ai > ai-1 and ai > ai+1. In words, i is awesome if the value at position i is larger than both its...
12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3.
12. Let D = {2E C | 너く1} denote the open unit disc and let f : D → C be a holomorphic function. Suppose that for any integer n>1 we have that f(1/n)-1/n3. Show that f(z)3.
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Let N denote a nonnegative integer-valued random variable. Show that k-1 k O In general show that if X is nonnegative with distribution F, then and E(X") = : nx"-'F(x) ds.
Let
be a permutation of {1,2,……n}.Let
-1 be the (n-1)-tuple with one element from
missing.
Alice shows Bob
-1[i] one by one in the increasing order of i from 1 to
(n-1).bob’s task is to compute the missing element from
-1 that is in
with very limited – O(log n) bits – of memory.
Design an algorithm to compute the missing element in this
memory-limited and access-limited model, i.e Alice can only show
each number to Bob once, and Bob...
1. Recall the following theorem. Theorem 1. Let a, b, m,n e N, m, n > 0 and ged(m,n) = 1. There erists a unique r e Zmn such that the following holds. x = a (mod m) x = b (mod n) please show that such solution is unique.
Exercise 1. Suppose we have a sample of values X1, X2, .-. , Xn. Let Yk denote the kth order statistics of this sample. Fix p € (0,1). Write (n+1)p=r+ for a unique integer r and d e [0,1). Show the following: (i) (# of sample values s yr) = r = np+(p-O). (ii) (# of sample values > yr) = n-r= n(1- p) - p+d.