Foundations of the theory of probability

42 IV.MathematicalExpectations

§

3. TheTchebycheffInequality

Let

f(x)

bea non-negativefunction ofa realargumentx,

whichforx

^

aneverbecomessmallerthanb

>

0.Thenforany-

randomvariablex

p[*^)s»,

(i)

provided themathematicalexpectation

E

{/(*)} exists.For,

E{f(x)}=jf(x)

P(dE)

^jf(x)P(dE)^bP(x^a)

,

fromwhich

(1)

followsatonce.

Forexample,

foreverypositive

c

,

P(x^a)^

E

-^. (2)

Nowlet

f(x)

benon-negative,even,and,forpositivex,non-

decreasing.Thenforeveryrandomvariablexandforanychoice

oftheconstanta>

thefollowinginequalityholds

P(|*|fea)3S

Iipp.

(3).

Inparticular,

P(|*-E(*)|

^

a)

£

E/{

Vf

W>

• 
  

(4)

1

f(a)

Especially

importantisthecase

f(x)

=

x

2

.Wethenobtainfrom

(3)and

(4)

P(\x\&*)^^p.

(5)

P(|,-E

W

|^.)^ife^.^,

(6)

where

oHx)

=E{x-E(x)}*

iscalledthevarianceofthevariablex.Itiseasytocalculatethat

o*(x)

=

E(x*)-{E(x)y.

If

f(x)

isbounded:

\f(x)

\^K,

thenalowerboundforP(\x\

^

a) canbefound.For