Page 265 - DMTH404_STATISTICS

Unit 18: The Weak Law

Notes
n
n (k n(p  

  e   ))   p q n k
k
k
k 0  

 k   n  n  n   q k  p n k
P   p    e    (pe ) (qe )

 n  k 0 k
  
q

= e  n   pe  qe  p  n ...(18.6)
2
x
Since e  x + e for any real x,
x
2 2 2 2
q
–p
pe + qe  p(q + e  q ) + q(–p + e  p )
2 2 2 2 2

= pe  q + qe  p  e . ...(18.7)
Substituting (18.7) into (18.6), we get
 k   2 n 
n

P   p    e .
 n 
But  n – n is minimum for  = /2 and hence
2
 k   n 2 /4

P   p    e ,   0. ...(18.8)
 n 
Similarly
 k   n 2 /4

P   p    £ e
 n 
and hence we obtain Bernstein’s inequality
 k  2
P   p    £ 2e  n /4 . ...(18.9)
 n 
Bernstein’s inequality is more powerful than Tchebyshev’s inequalityas it states that the chances
for the relative frequency k /n exceeding its probability p tends to zero exponentially fast as
n  .
Chebyshev’s inequality gives the probability of k /nto lie between and for a specific n. We can
use Bernstein’s inequality to estimate the probability for k /nto lie between and for all large n
Towards this, let

 k 
y   p     p   
n
 n 
so that

 n 

c  n 2 /4


P(y ) P   p    2e
n
 k 

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