StartComp18065: final project

#' @title a function to generate samples from a Rayleigh(σ) distribution
#' @description *Implement a function to generate samples from a Rayleigh(σ) distribution,using antithetic variables.
#'               What is the percent reduction in variance of (X+X)/2?
#'               compared with (X 1 +X 2)/2 for independent X 1 , X 2 ?
#' @param n The number of random Numbers generated
#' @param sigma Parameter of Rayleigh(σ) distribution
#' @param ant Parameter of function
#' @return a random sample of size \code{n}
#' @examples
#' \dontrun{
#' idpt_sample <-Rayleigh(4000,2,ant = F)#obtain 4000 independent samples
#' ant_sample<-Rayleigh(4000,2,ant = T)#4000 antithetic samples
#' var_idpt_sample<- var((idpt_sample[1:2000]+idpt_sample[2001:4000])/2)#compute the variance of independent sample
#' var_ant_sample <- var((ant_sample[1:2000]+ant_sample[2001:4000])/2)#compute the variance of antithetic sample
#' cat('the variance of independent sample is',var_idpt_sample,'.\n
#'     the variance of antithetic sample is',var_ant_sample,'.\n
#'     the percentage of variance reduction is',(var_idpt_sample-var_ant_sample)/var_idpt_sample,'.\n\n
#'     the covariance of independent sample is',(cov(idpt_sample[1:2000],ant_sample[2001:4000]))) # show the variance
#' }
#' @export
Rayleigh <- function(n,sigma,ant = T){
  m <- floor(n/2)
  u <- runif(m)
  if(ant) v<-1-u else v<- runif(m)
  u<-c(u,v)
  sample<-numeric(length(u))
  sample<-(-(2*sigma^2)*(log(1-u)))^(1/2)
  return(sample)
}