SECTION 6.3 Parameters and Statistics 377
6.3.3 The distribution ofX and the Central Limit Theorem
The result of this section is key to all of sampling theory. As we might
guess, one of the most important statistics we’re apt to encounter is
the mean x of n independent samples taken from some population.
Underlying this is the random variableXwith parameters
E(X) = μ, and Var(X) =σ^2
n.
Let’s start by getting our hands dirty.Simulation 1. Let’s take 100 samples of the mean (where each mean
is computed from 5 observations) from the uniform distribution having
density function
f(x) =
1 if 0≤x≤ 1 ,
0 otherwise.We display the corresponding histogram:
Simulation 2. Here, let’s take 100 samples of the mean (where each
mean is computed from 50 observations) from the uniform distribution
above. The resulting histogram is as below.