The Euler totient function : N → N is defined byo(n) = #{k ≤n: gdc(n, k) = 1}.That is, (n) is the count of all positive integer k that is relatively prime to n. E.g.,(1) = 1. Let X be a random number uniformly chosen from {1, 2,..., 20}, and Y =(a) Find the range of Y.(b) Find the probability mass function of Y. Present your answer in a table, e.g.,ke range(Y) Ppy (k)12:(9) = 6,(X).(c) Compute the expected value and variance of Y. (Hint: excel could be helpful.)

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Answered: The Euler totient function : N→ N is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let {Xn}n>1 be a sequence of random variables on some probability space such that (a) EX? 1; (b) E(X;X;) = E(X;)E(X;), i ± j; (c) C-1 % → 0, as n → co, -k%3D1°K where of = var(Xx). Then for X, = C, X;/n and ūn = CE(X;), show k X, – in as n → o. 2.
4 Let X₁, i=1,2,..., n be mutually uncorrelated random vectors of same size and with E(X;) = µ¡. Show that n E(|| Σ(X; – μ;)||2) = ΣΕ||X; – μ;||2 i=1 i=1 n
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