Page 268 - DMTH404_STATISTICS

Statistics

Notes
Figure 18.1

18.1 Summary

Let X be independent, identically distributed Bernoulli randomVariables such that
i
P(X ) = p, P(X = 0) = 1 – p = q,
i i
and let k = X + X + ... + X represent the number of “successes”in n trials. Then the weak law due
1 2 n
to Bernoulli states that [see Theorem 3-1, page 58, Text]
 k  pq
P   p    2 ...(18.1)

 h  n
i.e., the ratio “total number of successes to the total numberof trials” tends to p in probability as
nincreases.
A stronger version of this result due to Borel and Cantellistates that the above ratio k/n tends to
p not only in probability, but with probability 1. This is the strong law of large numbers (SLLN).

18.2 Keywords

Strong law of large numbers: A stronger version of this result due to Borel and Cantellistates
that the above ratio k/n tends to p not only in probability, but with probability 1. This is the
strong law of large numbers (SLLN).

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  .

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