Random Variables and Random Sampling-Distribution of X
Consider any population, call it P, whose characteristic we wish to estimate. There always
exists an associated derived population PD whose elements are all possible means of
subsets of size n (specified) from P, where subset selection is with replacement between
each draw. Suppose population P has mean μ and variance σ2.
To estimate characteristics of P we take a random sample of size n from P and compute
various summary statistics and graphs.
Let X1, X2,…, Xn denote sampling random variables which mathematically describe the
sample from P. We know EXi = μ and VAR Xi = σ 2, for each i = 1, 2, …, n. The sample
n
1
mean random variable is X = ∑ X i .
n i =1
X is defined on the derived population PD. We know E X = μ and VAR X = σ 2/n. I.e.
PD has mean μ and variance σ 2/n, for specified sample size n.
The following facts are known about the distribution of X (I.E. about population PD):
1.
If P is a normal (μ, σ 2) population, then PD is a normal (μ, σ 2/n) population for any
n ≥ 1.
2.
If P is any population, and n is large (n ≥ 25 say) the PD is approximately a normal
(μ, σ 2/n) population, (CLT).