The following is a sufficient condition, the Laplace–Liapounoff condition, for the central limit theorem: If X1, X2, X3, ... is a sequence of independent random vari-ables, each having an absolute third moment ci = E(|Xi − μi|3) and if lim n→q[var(Yn)]− 32 ·ni=1ci = 0 where Yn = X1 + X2 +···+ Xn, then the distribution of the standardized mean of the Xi approaches the stan-dard normal distribution when n→q. Use this condi-tion to show that the central limit theorem holds for the sequence of random variables of Exercise 7.
The following is a sufficient condition, the Laplace–Liapounoff condition, for the central limit theorem: If X1, X2, X3, ... is a sequence of independent random vari-ables, each having an absolute third moment ci = E(|Xi − μi|3) and if lim n→q[var(Yn)]− 32 ·ni=1ci = 0 where Yn = X1 + X2 +···+ Xn, then the distribution of the standardized mean of the Xi approaches the stan-dard normal distribution when n→q. Use this condi-tion to show that the central limit theorem holds for the sequence of random variables of Exercise 7.
Chapter9: Sequences, Probability And Counting Theory
Section9.1: Sequences And Their Notations
Problem 71SE: Prove the conjecture made in the preceding exercise.
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The following is a sufficient condition, the Laplace–
Liapounoff condition, for the central limit theorem: If
Liapounoff condition, for the central limit theorem: If
X1, X2, X3, ... is a sequence of independent random vari-
ables, each having an absolute third moment
ables, each having an absolute third moment
ci = E(|Xi − μi|
3)
3)
and if
lim n→q[var(Yn)]
− 3
2 ·
n
i=1
ci = 0
− 3
2 ·
n
i=1
ci = 0
where Yn = X1 + X2 +···+ Xn, then the distribution
of the standardized mean of the Xi approaches the stan-
dardnormal distribution when n→q. Use this condi-
tion to show that the central limit theorem holds for the
dard
tion to show that the central limit theorem holds for the
sequence of random variables of Exercise 7.
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