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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

Compute the measurement signal-to-noise ratio that is, |渭|/蟽, where 渭 = E[X] and 蟽2 = Var(X) of the

following random variables:

(a) Poisson with mean 位;

(b) binomial with parameters n and p;

(c) geometric with mean 1/p;

(d) uniform over (a, b);

(e) exponential with mean 1/位;

(f) normal with parameters 渭, 蟽2.

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

A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the number of fish that need be caught to obtain at least one of each type.

(a) Give an interval (a, b) such thatP(a鈮�Y鈮�b)鈮�0.9

(b) Using the one-sided Chebyshev inequality, how many fish need we plan on catching so as to be at least 90 percent certain of obtaining at least one of each type?

Each new book donated to a library must be processed. Suppose that the time it takes to process a book has a mean of 10minutes and a standard deviation of 3minutes. If a librarian has 40 books to process,

(a) approximate the probability that it will take more than 420minutes to process all these books;

(b) approximate the probability that at least 25 books will be processed in the first 240 minutes. What assumptions have you made?

Use the central limit theorem to solve part (c)of the problemlocalid="1649757874152" 8.2.
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