Results 1 to 2 of 2

Math Help - Central Limit Theorem

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    154

    Central Limit Theorem

    Let  X_{1}, X_{2}, \ldots, X_{n} be a random sample from a distribution with mean  \mu and variance  \sigma^{2} . Then, in the limit as  n \to \infty , the standardized versions of  \bar{X} and  T_0 have the standard normal distribution. That is  \lim_{n \to \infty} P \left (\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \leq z \right) = P(Z \leq z) = \Phi(z) and  \lim_{n \to \infty} P \left (\frac{T_0 - n\mu}{\sqrt{n}\sigma} \leq z \right) = P(Z \leq z) = \Phi(z) .

    My question is, why should this be the case? Why shouldn't  \bar{X} and  T_{0} have a lognormal distribution, an exponential distribution, etc..?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by heathrowjohnny View Post
    Let  X_{1}, X_{2}, \ldots, X_{n} be a random sample from a distribution with mean  \mu and variance  \sigma^{2} . Then, in the limit as  n \to \infty , the standardized versions of  \bar{X} and  T_0 have the standard normal distribution. That is  \lim_{n \to \infty} P \left (\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \leq z \right) = P(Z \leq z) = \Phi(z) and  \lim_{n \to \infty} P \left (\frac{T_0 - n\mu}{\sqrt{n}\sigma} \leq z \right) = P(Z \leq z) = \Phi(z) .

    My question is, why should this be the case? Why shouldn't  \bar{X} and  T_{0} have a lognormal distribution, an exponential distribution, etc..?
    Well for one thing these other distributions don't have the property that
    the mean of two independent identically distributed RV's don't have the same
    distribution (but with half the variance of either). Any limiting distribution for
    mean must have this property.

    Essentially you are looking for stable distributions, once you have these you
    will find that there is a generalised central limit theorem in which other stable
    distributions may appear.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: July 21st 2010, 04:27 PM
  2. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 17th 2010, 06:42 PM
  3. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: June 17th 2009, 10:27 PM
  4. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 4th 2008, 12:36 AM
  5. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: October 21st 2007, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum