Sample Mean

1
Chapter 10
2
Sample Mean

• Definition Sample Mean
• Assume EXi m for all i, then
• EMn m
• Hence, Mn is an unbiased estimate of m.
• If Var(Xi) ?2 for all i, then
• Convergence in probability

• Since Mn converges to m in probability, it is a
  consistent estimator of m.
• It can be shown that
• Var(Mn) ?2/n
• Thus one may work on a normalized random variable
  
• Clearly, EYn 0, Var(Yn) 1

3
Sample Mean (Contd)

• Let the CDF of ZnnMn be Gn(z). Then
• its cdf
• Example 11.1 If Xi N(m, ?2), then nMn N(nm,
  n?2). Let ?(x) be the cdf of the standard
  uni-variate normal density N(0,1). Then
• and Fn(y) ?(y). Note that Fn(y) is independent
  of n. If Xi is not normal, however, Fn(y) will be
  dependent on n.

• Example 11.2 Recall that
• Substitute Yn, we have
• Let Then
• If Xis N(0,1), and ? ?,
• Above bound is loose. For example, if n 3, then

4
Central Limit Theorem

• Theorem (CLT) Let X1, X2, , Xn be iid random
  variables with finite mean m and variance ?2.
  Denote
• Then
• Significance of CLT
• Even is unknown,
• can be
  approximated by for n gt 30.

• Example 11.3 A channel has bit error probability
  p. Use CLT to estimate P(more than k bits are
  incorrect when n bits are transmitted)
• Solution Denote Xi 1 if bit i is received in
  error, 0 if correct. Then Xi Bernoulli(p).
  Thus, EXi p, Var(Xi) p(p-1). Hence

5
Confidence Interval

• Since
• using CLT, a 100(1??)
  confidence interval of m can be specified to
  choose appropriate value of y such that
• where ? is determined by user.
• To determine y, solve y?/2 s.t.
• The length of
• is

• Example 11.5 Let X1, X2, , Xn be i.i.d. with
  variance ?2 2. If M100 7.129. Find the 93
  confidence interval of EXi m.
• Solution Look-up in table 11.1.
• 1?? 0.93, y?/2 1.812. The confidence interval
  is
• In other words,
• M 7.129 ? 0.2563 with 93 probability.

6
Sample Variance

• In calculating confidence interval, we assume the
  variance ?2 is known. When ?2 is unknown, we need
  to estimate the sample variance
• a) ES2 ?2 unbiased est.
• b) consistent est.

• Given the sample variance, during estimation of
  the confidence interval of m, we use S2 in lieu
  of ?2. The remaining steps are still the same.