1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)
Question2: 1. f(n)-O(g(n) if there exist c, no>0 such that f(n)for all n 2 no- 2. f(n)-2(g(n)) if there existc, no>0 such that f(n)for all n 2 no- 3. f(n)- (g(n)) if there exist C1, C2,no > 0 such that-for all n 2 no-
Find the Fourier series of the following function, and calculate the sum of rn. n=1 f(x) = 12,2 if 0<r<\ if-1< 0 f(x + 2)-f(x)
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F). Set Cn = An ñ Bn, Dn = An U Bn: (1) Show that (Tim An) ^ ( lim Bm) – lim Cn (lim An) ( lim Bu) C lim Dm and 100 noo (2) Show by example the two inclusions in (1) can be strict.
Problem 1 (15%): Find the following probabilities for two normal random variables Z = N(0,1) and X = N(-1,9). (a) P(Z > -1.48). (b) P(|X< 2.30) (c) What is the type and the parameters of the random variable Y = 3X +5?
10. Let a, b,n E Z such that n >0, n does not divide a and al B in Z/nZ. Assume a-and [N]-[a]. Prove n #313 and n 497, 4
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
' cos(3t), t<n/2, 2. Let f(t) = sin(2t), 7/2<t< , Write f(t) in terms of the unit step e3 St. function. Then find c{f(t)}.
Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0 (1, M(n) = {3}: M(n − 1) – 2n +1, if n > 0 f(n) = n +1 Prove that M(n) = f(n) for all n > 0.