1
251cheby 10/4/04
Proof and Extensions of Chebyshev Inequality
The following is a retelling of work presented by Henry Bottomly (See
http://www.btinternet.com/~se16/hgb/cheb.htm#OTProof ,
http://www.btinternet.com/~se16/hgb/cheb2.htm and
http://www.mcdowella.demon.co.uk/Chebyshev.html
The Chebyshef inequality is a simple extension of Markov’s Inequality, which, in turn is based on the
definition of the mean.
E x 
.
a
Markov’s Inequality: If x takes only nonnegative values and a is any positive number, Px  a  
Proof: The definition of the mean says   E x  

xPx  
a 1

xPx  
0

 xPx . Here what we are
a
saying is that we can stop our summing of probabilities at any point we wish, and then resume. Since the
a 1
part of this sum is below a is nonnegative

xPx   0 , we can write   E x  

xPx  
0

 xPx .
a
But since all values of x in the last summation are  a ,



a
a
a
 xPx   aPx  a Px  a Px  a .
We have now shown   Ex   a Px  a  . and since a  0, we can divide the inequality by a to get
E x 
E x 
 Px  a  or Px  a  
.
a
a


1
Chebyschef Inequality: This can be written P x    k 
Proof: The Markov inequality says P y  a  
Then

P x
or P  k  x    k   1 
Ey
. Let y  x  
a

2 and a  t 2 .
 E  xt    . But   Ex      E x   . This means that

  t   t But if t is positive, to say  x     t is the same as saying  x     t .
P x     t2 
2
2
2
k
2
2
2
2
2
2
So we can write P  x     t  
2
t2
.
Now let t  k , where k  0 . Then P x     k  
251cheby 10/4/04
2
2
2
2
2
1
or P x     k   2 .
2 2
k 
k
1
k2
2
The one-tailed version of this is Px     k  
Px     k  
1
1
calculus.
t2
2

2
 2 t2
1
1 k 2
, which can also be written
. I am still trying to find a proof of this that is simple and does not use