Consider the sequence of independent random variables X1, X2, X3, … having the uniform densities
Use the sufficient condition of Exercise 7 to show that the central limit theorem holds.
The following is a sufficient condition for the central limit theorem: If the random variables X1, X2, … , Xn are independent and uniformly bounded (that is, there exists a positive constant k such that the probability is zero that any one of the random variables Xi will take on a value greater than k or less than âˆ’k), then if the variance of
becomes infinite when nâ†’q, the distribution of the standardized mean of the Xi approaches the standard normal distribution. Show that this sufficient condition holds for a sequence of independent random variables Xi having the respective probability distributions