The Euler totient function : N→ N is defined by o(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., (9) = 6, (1) = 1. Let X be a random number uniformly chosen from {1,2,..., 20}, and Y = o(X). (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) Py (k) 1 2 ⠀ (c) Compute the expected value and variance of Y. (Hint: excel could be helpful.)
The Euler totient function : N→ N is defined by o(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., (9) = 6, (1) = 1. Let X be a random number uniformly chosen from {1,2,..., 20}, and Y = o(X). (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) Py (k) 1 2 ⠀ (c) Compute the expected value and variance of Y. (Hint: excel could be helpful.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:The Euler totient function : N → N is defined by
o(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)
1
2
:
(9) = 6,
(X).
(c) Compute the expected value and variance of Y. (Hint: excel could be helpful.)
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