Central Limit Theorem

It is possible to illustrate the Central Limit Theorem empirically. It is also possible to illustrate the Central Limit Theorem theoretically. The following examples show how this can be done.

1. [Exercise 5.4.4 and Figure 5.5 in Hogg/Tanis] Let [Maple Math] , ..., [Maple Math] be a random sample of size n from a U -shaped distribution with p.d.f.

f(x) = [Maple Math] , [Maple Math] < x < 1. Empirically show that [Maple Math] is approximately N (0, 1) when n is sufficiently large. Solution

2. [Exercise 5.4.10 and Figure 5.5 in Hogg/Tanis] Let [Maple Math] , ..., [Maple Math] be a random sample of size n from a U -shaped distribution with p.d.f. [Maple Math] , [Maple Math] < x < 1. Theoretically show that the p.d.f. of [Maple Math] = [Maple Math] becomes closer and closer to the N(0,1) p.d.f. as n increases. Solution

3. [Exercise 5.4.11(a)] Let [Maple Math] , ..., [Maple Math] be a random sample of size n from the uniform distribution on the interval [ [Maple Math] , 1]. I.e., the distribution is U ( [Maple Math] , 1). Show that the p.d.f. of [Maple Math] = [Maple Math] becomes closer and closer to the N (0, 1) p.d.f. as n increases.

4. [Exercise 5.4.11(b) and Figure 5.4 of Hogg/Tanis] Let [Maple Math] , ..., [Maple Math] be a random sample of size n from a triangular distribution on the interval [ [Maple Math] , 1] with p.d.f. defined by f(x) = [Maple Math] , [Maple Math] < x < 1. For this distribution, [Maple Math] , [Maple Math] . Show that the p.d.f. of [Maple Math] = [Maple Math] becomes closer and closer to the N (0, 1) p.d.f. as n increases.