EBK INTRODUCTION TO THE PRACTICE OF STA
8th Edition
ISBN: 9781319116828
Author: Moore
Publisher: VST
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Chapter 4.4, Problem 85E
To determine
To find: The effect if the people having high intake of calcium are more compliant on the overall standard deviation.
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6. Let, for p = (0, 1), and xe R. X be a random variable defined as follows:
P(X=-x) = P(X = x)=p. P(X=0)= 1-2p.
Show that there is equality in Chebyshev's inequality for X.
This means that Chebyshev's inequality, in spite of being rather crude, cannot be
improved without additional assumptions.
4. Prove that, for any random variable X, the minimum of EIX-al is attained for
a = med (X).
8. Recall, from Sect. 2.16.4, the likelihood ratio statistic, Ln, which was defined
as a product of independent, identically distributed random variables with mean
1 (under the so-called null hypothesis), and the, sometimes more convenient,
log-likelihood, log L, which was a sum of independent, identically distributed
random variables, which, however, do not have mean log 1 = 0.
(a) Verify that the last claim is correct, by proving the more general statement,
namely that, if Y is a non-negative random variable with finite mean, then
E(log Y) log(EY).
(b) Prove that, in fact, there is strict inequality:
E(log Y) < log(EY),
unless Y is degenerate.
(c) Review the proof of Jensen's inequality, Theorem 5.1. Generalize with a
glimpse on (b).
Chapter 4 Solutions
EBK INTRODUCTION TO THE PRACTICE OF STA
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