Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

82 Probability and Distributions

Example 1.10.3.LetXbe a nondegenerate random variable with meanμand a
finite second moment. Thenμ^2 <E(X^2 ). This is obtained by Jensen’s inequality
using the strictly convex functionφ(t)=t^2.

The last inequality concerns different means of finite sets of positive numbers.

Example 1.10.4(Harmonic and Geometric Means). Let{a 1 ,...,an}be a set of
positive numbers. Create a distribution for a random variableXby placing weight
1 /non each of the numbersa 1 ,...,an. Then the mean ofXis thearithmetic
mean,(AM),E(X)=n−^1

∑n
i=1ai. Then, since−logxis a convex function, we
have by Jensen’s inequality that

−log

(

1

n

∑n

i=1

ai

)

≤E(−logX)=−

1

n

∑n

i=1

logai=−log(a 1 a 2 ···an)^1 /n

or, equivalently,

log

(

1

n

∑n

i=1

ai

)

≥log(a 1 a 2 ···an)^1 /n,

and, hence,

(a 1 a 2 ···an)^1 /n≤

1

n

∑n

i=1

ai. (1.10.6)

The quantity on the left side of this inequality is called thegeometric mean(GM).
So (1.10.6) is equivalent to saying that GM≤AM for any finite set of positive
numbers.
Now in (1.10.6) replaceaiby 1/ai(which is also positive). We then obtain

1

n

∑n

i=1

1

ai

≥

(

1

a 1

1

a 2

···

1

an

) 1 /n

or, equivalently,

1

1

n

∑n

i=1

1

ai

≤(a 1 a 2 ···an)^1 /n. (1.10.7)

The left member of this inequality is called theharmonic mean(HM). Putting
(1.10.6) and (1.10.7) together, we have shown the relationship

HM≤GM≤AM, (1.10.8)

for any finite set of positive numbers.

EXERCISES
1.10.1.LetX be a random variable with meanμand letE[(X−μ)^2 k]exist.
Show, withd>0, thatP(|X−μ|≥d)≤E[(X−μ)^2 k]/d^2 k. This is essentially
Chebyshev’s inequality whenk= 1. The fact that this holds for allk=1, 2 , 3 ,...,
when those (2k)th moments exist, usually provides a much smaller upper bound for
P(|X−μ|≥d) than does Chebyshev’s result.