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Chapter 8: Q. 8.2 (page 392)

ItXhas, a mean渭and standard deviation蟽, the ratior=|渭|/蟽is called the measurement signal-to-noise ratioX. The idea is that Xcan be expressed asX=渭+(X鈭�渭), 渭representing the signal and X鈭�渭the noise. If we define|(X鈭�渭)/渭|=Dit as the relative deviation Xfrom its signal (or mean)渭, show that for伪>0,

P{D鈮�伪}鈮�1鈭�1r2伪2.

P{D鈮�伪}鈮�1鈭�1r2伪2

Short Answer

Expert verified

Since |渭|伪>0using Chebyshev's inequality we get:

P{|X-渭|>|渭|伪}鈮�蟽2渭2伪2.

Therefore,

P{D鈮�伪}=P{|X-渭|鈮�|渭|伪}鈮�1-蟽2渭2伪2=r=|u|=1-1r2伪2.

Step by step solution

01

Given Information.

Xhas to mean渭and standard deviation蟽, the ratiolocalid="1649850029051" r=|渭|/蟽called the measurement signal-to-noise ratio of X.

02

Explanation.

Assume that the random variable Xhas mean 渭and standard deviation蟽, and let伪>0. Then,

P{D鈮�伪}=P鈭�X-渭渭鈮�伪=P{|X-渭|鈮�|渭|伪}

Since localid="1649850016019" |渭|伪>0using Chebyshev's inequality we get:

P{|X-渭|>|渭|伪}鈮�蟽2渭2伪2.

Since

P{|X-渭|>|渭|伪}=1-P{|X-渭|鈮�|渭|伪},

therefore

P{D鈮�伪}=P{|X-渭|鈮�|渭|伪}鈮�1-蟽2渭2伪2鈬�r=|渭|蟽P{D鈮�伪}鈮�1-1r2伪2.

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Most popular questions from this chapter

The Chernoff bound on a standard normal random variableZgivesP{Z>a}鈮�e-a2/2,a>0. Show, by considering the densityZ, that the right side of the inequality can be reduced by the factor2. That is, show that

P{Z>a}鈮�12e-a2/2a>0

Suppose that a fair die is rolled 100times. Let Xibe the value obtained on the ith roll. Compute an approximation forP鈭�1100Xi鈮�a1001<a<6.

Let X1, ... , X20 be independent Poisson random variables with mean 1.

(a) Use the Markov inequality to obtain a bound on

P鈭�Xi>15120

(b) Use the central limit theorem to approximate

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A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least 0.95?

The strong law of large numbers states that with probability 1, the successive arithmetic averages of a sequence of independent and identically distributed random variables converge to their common mean . What do the successive geometric averages converge to? That is, what is

limn鈫�鈭�鈭�i=1nXi1/n
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