From Jonathan Yam: The Central limit Theorem states that when sample size tends to infinity, the sample mean will be normally distributed. The Law of Large Number states that when sample size tends to infinity, the sample mean equals to population mean. Is the two statements contradictory? Answered by Paul Betts and Harley Weston.
From Donna Hall: A irregularly shaped object of unknown area A is located in the unit square 0<=x<=1. Consider a random point uniformly distributed over the square. Let X = 1 if the point lies inside the object and X = 0 otherwise. Show that E(X) = A. How could A be estimated from a sequence of n independent points uniformly distributed over the square? How would you use the central limit theorem to gauge the probable size of the error of the estimate. Answered by Harley Weston.
From Donna Hall: A skeptic gives the following argument to show that there must be a flaw in the central limit theorem: We know that the sum of independent Poisson random variables follows a Poisson distribution with aparameter that is the sum of the parameters of the summands. In particular, if n independentPoisson random variables, each with parameter 1/n, are summed, the sum has a Poisson distributionwith parameter 1. The central limit theoren says the sum tends to a normal distribution, butPoisson distribution with parameter 1 is not normal.
From Donna D.Hall: I am looking for a proof for the normal distribution.
I suppose "proof" was not a good choice of words. What I am looking for is a way to "derive" the normal distribution in simple terms so that the most average teenager can see the logic. Can you help me?
Answered by Harley Weston.
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.