############################################################################################################
# This program calculates the variance of a mean when the units that are included in it are chosen at random
# from a finite population of $1, ..., N$ units.
#
# The program is used to calculate the expresion: \frac{1}{m} \cdot \sum_{i=1}^N Y_i(0) \cdot T_i
# The program is written such that X is equal to Y(0), however we can equally define X to be Y(1) and then we will need to switch "tr" with "1-tr"
############################################################################################################
meanBernulli = function(sigma0,mean0,n, p){
stopifnot( length(c(sigma0,mean0,n, p))==4 )
var1 = n * p * (1-p) * ( sigma0^2 + mean0^2 )
var2 = n * p * (1-p)
covariance = p * (1-p) * n * mean0
Sigma = matrix(c(var1, covariance, covariance, var2), byrow=TRUE, ncol=2 )
A = matrix(c( 1/(n*p) , - p * n * mean0/(n*p)^2 ), byrow=TRUE, ncol=1 )
variance.of.mean = t(A) %*% Sigma %*% A
return(variance.of.mean)
}
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