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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).