Question 2. (exercise 2.16 in textbook) Let A E A, for i 1,2,...,n, be a sequence...
Question 2. (exercise 2.16 in textbook) Let A E A, for i 1,2,...,n, be a sequence of events. Show that
I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n exists a C > 0 such that for n 1,2, 3,.. Show that X is cons istent for ?
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
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2. Let X, N o ?) for i=1,2. Show that Y = X1 + X, and Z X; - X2 are independent. 3. Let 2-N(0,1) and W x (n) with Z be independent of W. Show that the distribution of T- tudiatvihustion with n deerees of freedom. (Hint: create a second variable U - find the joint distribution
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e) divergence at I = -5? Exercise 7.10. Show that if the sequence and is bounded then the power series > .7 n=0 converges absolutely for p<1. Exercise 7.11. Let A be a set of real numbers with the following property: For every real number Il i) if I, E A then I e A for every I such that I< 1:1), and ii) if I & A then I ¢ A for every I such that :|...
An array A[1,2,... ,n is unimodal if its consists of an increasing sequence followed by sequence a decreasing sequence. More precisely, there exists an index k є {1,2,… ,n} such that there exists an indes . AlE]< Ali1 for all 1 i< k, and Ai]Ali 1 for all k< i< n A1,2,..,n] in O(logn) time the loop invariant (s) that your algorithm maintains and show why they lead to the correctness Give an algorithm to compute the maximum element of...
2. Let 'n ,n > l be a sequence of r.v.s such that E[Xi] μί and Var(X) σ for i-: 1, 2, , and Cov(Xi, Χ.j) Ơij for i J. Let {an ,n 1) and (bn, n 1) be the sequences of real numbers. Write down the expressions for i-l (i,Xi, Xi), Cov every i and Ơij 0 for every i j, state Var(Σί ! així), Coy(Σ, aixi, xi),
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,...
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.)
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
Question 1. (exercise 26 in textbook) Let A be a σ algebra of subsets of Ω and let B E A Show that F = {An B : A e A} is a σ algebra of subsets of B Is it still true when B is a subset of Ω that does not belong to A?