平均數及變異數的抽樣分配(Distribution of Mean and Variance)

為什麼這些估計值會是隨機變數呢?

Julia程式碼

using Random, Distributions, Plots; pyplot()
Random.seed!(0)
# 假設母體分配為exponetial distribution
lambda  = 1/4.5
expDist = Exponential(1/lambda)
# n為每一次抽樣的樣本大小
# N為抽樣次數
n, N    = 10, 10^6
means     = Array{Float64}(undef, N)
variances = Array{Float64}(undef, N)
# 開始進行模擬抽樣，重複N次
for i in 1:N
    # 隨機(模擬)抽取樣本資料
    data = rand(expDist,n)
    # 計算樣本平均及變異數
    means[i] = mean(data)
    variances[i] = var(data)
end
# 理論的平均和抽樣得出的樣本平均
println("Actual mean: ",mean(expDist),
        "\nMean of sample means: ",mean(means))
println("Actual variance: ",var(expDist),
    "\nMean of sample variances: ",mean(variances))
# 用直方圖表示其分配
stephist(means, bins=200, c=:blue, normed=true,
    label="Histogram of Sample Means")
stephist!(variances, bins=600, c=:red, normed=true,
    label="Histogram of Sample Variances", xlims=(0,40), ylims=(0,0.4),
        xlabel = "Statistic value", ylabel = "Density")