Empirical Examples of the Central Limit Theorem

1
Empirical Examples of the Central Limit Theorem

• Lecture XVI

2
Back to Asymptotic Normality

• The characteristic function of a random variable
  X is defined as
• Note that this definition parallels the
  definition of the moment-generating function

3

• Like the moment-generating function there is a
  one-to-one correspondence between the
  characteristic function and the distribution of
  random variable.
• Two random variables with the same characteristic
  function are distributed the same.

4

• The characteristic function of the uniform
  distribution function of the uniform distribution
  function is
• The characteristic function of the Normal
  distribution function is

5

• The Gamma distribution function
• which implies the characteristic function

6

• Taking a Taylor series expansion of around the
  point t 0 yields

7

• To work with this expression we note that
• For any random variable X, and

8

• Putting these two results into the second-order
  Taylor series expansion

9

• Thus

10
Wrapping Up Loose Ends.

• Application of Holders Inequality.
• Holders Inequality
• Example 4.7.1 If X and Y have means mX, mY and
  variances sX2, sY2, respectively, we can apply
  the Cauchy-Schwartz Inequality (Holders
  inequality with pq1/2) to get

11

• Squaring both sides and substituting for
  variances and covariances yields
• Which implies that the absolute value of the
  correlation coefficient is less than one.

12
Application of Chebychevs Inequality

• Chebychevs Inequality Let X be a random
  variable and let g(x) be a nonnegative function.
  Then, for any rgt0

13

• Example 4.7.3 The most widespread use of
  Chebychevs Inequality involves means and
  variances. Let g(x)(x-m)2/s2, where mE X
  and s2V (X ). Let rt2.

14
(No Transcript)
15

• Letting t2, this becomes
• However, this inequality may not say much, since
  for the normal distribution

16

• Casella and Berger offer a proof of the Central
  Limit Theorem (Theorem 5.3.3) based on the moment
  generating function instead of the characteristic
  function. However, they note that the proof
  using the characteristic function is a stronger
  result.

17
Normal Approximation of the Binomial

• Starting from the Binomial distribution function

18

• First, assume that n10 and p.5. The
  probability of r ? 3 is
• Note that this distribution has a mean of 5 and a
  variance of 2.5. Given this we can compute

19

• Integrating the standard normal distribution
  function from negative infinity to 1.265 yields

20

• Expanding the sample size to 20 and examining the
  probability that r ? 6 yields
• This time the mean of the distribution is 10 and
  the variance is 5. The resulting z-1.7889.
  PZ.0368.

21

• As the sample size declines the binomial
  probability approaches the normal probability.