The estimator is
the sample mean: 1n sum x_i#
We claim that as the size of sample increase, the sample mean
converge website here the population mean#
the following is the sample size vector contains 24 candidiate sample
sizeSample_Size
= 2^(0:23)#
initialize a vector with 24 entries to store the sample mean value
with each sample size candidiateSample_Mean
= numeric(length = length(Sample_Size))#
calculate the sample mean for each sample size candidiatefor
(i in 1:length(Sample_Size))
Sample_X
= sample(X, size = Sample_Size[i] , replace = FALSE, prob = NULL)
Mean_X
= mean(Sample_X)
Sample_Mean[i]=Mean_X#
plot the sample mean for each sample size candidiateplot(Sample_Size,
Sample_Mean, log = “x”, ylim =c(E-sd,E+sd),
xlab
=’Sample Size’, ylab = ‘Sample Mean’, col = ‘steelblue’,
main
= ‘Sample Mean Converge to Population Mean’,cex. This is a statistically proven fact.
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The weak law of large numbers (also called Khinchin’s law) states that the sample average converges in probability towards the expected value17
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