(a) Let X be a discrete random variable taking non-negative values; i.e. Rx = No = {0, 1, 2, ...}. Let a > 0. Show that E[X] > aP(X > a) and deduce that P(X > a) < E[X]/a, an inequality known as Markov's inequality. (b) Let X be a discrete random variable. By applying Markov's inequality to a well-chosen random variable, prove Chebyshev's inequality; that is, show that if a > 0 then Var[X] P (|X – EX|> a)< (c) Suppose that F ~ Bin(2m + 1, 5). Use Chebyshev's inequality to bound P[F > m]. From (c), deduce that the probability of a decoding error in the binary sym- metric channel (BSC) with symbol error probability p = can be made arbitrarily small by using a sufficiently long repetition code. сan
(a) Let X be a discrete random variable taking non-negative values; i.e. Rx = No = {0, 1, 2, ...}. Let a > 0. Show that E[X] > aP(X > a) and deduce that P(X > a) < E[X]/a, an inequality known as Markov's inequality. (b) Let X be a discrete random variable. By applying Markov's inequality to a well-chosen random variable, prove Chebyshev's inequality; that is, show that if a > 0 then Var[X] P (|X – EX|> a)< (c) Suppose that F ~ Bin(2m + 1, 5). Use Chebyshev's inequality to bound P[F > m]. From (c), deduce that the probability of a decoding error in the binary sym- metric channel (BSC) with symbol error probability p = can be made arbitrarily small by using a sufficiently long repetition code. сan
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
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![(a) Let X be a discrete random variable taking non-negative values; i.e. Rx =
No = {0, 1, 2, ...}. Let a > 0. Show that E[X] 2 aP(X > a) and deduce
that P(X > a) < E[X]/a, an inequality known as Markov's inequality.
(b) Let X be a discrete random variable. By applying Markov's inequality to
a well-chosen random variable, prove Chebyshev's inequality; that is, show
that if a > 0 then
Var[X]
a²
P (|X – EX| > a) <
(c) Suppose that F ~ Bin(2m + 1, n). Use Chebyshev's inequality to bound
P[F > m].
* 10
From (c), deduce that the probability of a decoding error in the binary sym-
metric channel (BSC) with symbol error probability p = 1 can be made
arbitrarily small by using a sufficiently long repetition code.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff258f736-2efe-4abc-a546-b1bd73d984a7%2F26cf4384-7725-4ba2-8a54-271d3ae7d86e%2Fq18771_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let X be a discrete random variable taking non-negative values; i.e. Rx =
No = {0, 1, 2, ...}. Let a > 0. Show that E[X] 2 aP(X > a) and deduce
that P(X > a) < E[X]/a, an inequality known as Markov's inequality.
(b) Let X be a discrete random variable. By applying Markov's inequality to
a well-chosen random variable, prove Chebyshev's inequality; that is, show
that if a > 0 then
Var[X]
a²
P (|X – EX| > a) <
(c) Suppose that F ~ Bin(2m + 1, n). Use Chebyshev's inequality to bound
P[F > m].
* 10
From (c), deduce that the probability of a decoding error in the binary sym-
metric channel (BSC) with symbol error probability p = 1 can be made
arbitrarily small by using a sufficiently long repetition code.
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