# solution

The following is a sufficient condition, the Laplaceâ€“ Liapounoff condition, for the central limit theorem: If X1, X2, X3, … is a sequence of independent random variables, each having an absolute third moment

where Yn = X1 + X2 +Â·Â·Â·+ Xn, then the distribution of the standardized mean of the Xi approaches the standard normal distribution when nâ†’q. Use this condition to show that the central limit theorem holds for the sequence of random variables of Exercise 7.

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.

Exercise 7

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