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```
#' Demonstration of the Central Limit Theorem
#'
#' First of all, a number of \code{obs} observations are generated from a
#' certain distribution for each variable \eqn{X_j}, \eqn{j = 1, 2, \cdots, n}{j
#' = 1, 2, ..., n}, and \eqn{n = 1, 2, \cdots, nmax}{n = 1, 2, ..., nmax}, then
#' the sample means are computed, and at last the density of these sample means
#' is plotted as the sample size \eqn{n} increases (the theoretical limiting
#' distribution is denoted by the dashed line), besides, the P-values from the
#' normality test \code{\link{shapiro.test}} are computed for each \eqn{n} and
#' plotted at the same time.
#'
#' As long as the conditions of the Central Limit Theorem (CLT) are satisfied,
#' the distribution of the sample mean will be approximate to the Normal
#' distribution when the sample size \code{n} is large enough, no matter what is
#' the original distribution. The largest sample size is defined by \code{nmax}
#' in \code{\link{ani.options}}.
#'
#' @param obs the number of sample means to be generated from the distribution
#' based on a given sample size \eqn{n}; these sample mean values will be used
#' to create the histogram
#' @param FUN the function to generate \code{n} random numbers from a certain
#' distribution
#' @param mean,sd the expectation and standard deviation of the population
#' distribution (they will be used to plot the density curve of the
#' theoretical Normal distribution with mean equal to \code{mean} and sd equal
#' to \eqn{sd/\sqrt{n}}; if any of them is \code{NA}, the density curve will
#' be suppressed)
#' @param col a vector of length 4 specifying the colors of the histogram, the
#' density curve of the sample mean, the theoretical density cuve and
#' P-values.
#' @param mat,widths,heights arguments passed to \code{\link{layout}} to set the
#' layout of the two graphs.
#' @param xlim the x-axis limit for the histogram (it has a default value if not
#' specified)
#' @param \dots other arguments passed to \code{\link{plot.default}} to plot the
#' P-values
#' @return A data frame of P-values.
#' @author Yihui Xie
#' @references Examples at \url{https://yihui.org/animation/example/clt-ani/}
#' @seealso \code{\link{hist}}, \code{\link{density}}
#' @export
clt.ani = function(
obs = 300, FUN = rexp, mean = 1, sd = 1, col = c('bisque', 'red', 'blue', 'black'),
mat = matrix(1:2, 2), widths = rep(1, ncol(mat)), heights = rep(1, nrow(mat)), xlim, ...
) {
nmax = ani.options('nmax')
x = matrix(nrow = nmax, ncol = obs)
for (i in 1:nmax) x[i, ] = apply(matrix(replicate(obs, FUN(i)), i), 2, mean)
pvalue = apply(x, 1, function(xx) shapiro.test(xx)$p.value)
layout(mat, widths, heights)
if (missing(xlim)) xlim = quantile(x, c(.005, .995))
if (is.null(mean)) mean = NA
if (is.null(sd)) sd = NA
for (i in 1:nmax) {
dev.hold()
hist(x[i, ], freq = FALSE, main = '',
xlab = substitute(italic(bar(x)[i]), list(i = i)), col = col[1], xlim = xlim)
lines(density(x[i, ]), col = col[2])
if (!is.na(mean) && !is.na(sd))
curve(dnorm(x, mean, sd/sqrt(i)), col = col[3], lty = 2, add = TRUE)
legend('topright', legend = paste('P-value:', sprintf('%.03f', pvalue[i])), bty = 'n')
plot(pvalue[1:i], xlim = c(1, nmax), ylim = c(0, 1), xlab = 'n',
ylab = 'P-value', col = col[4], ...)
ani.pause()
}
invisible(data.frame(n = 1:nmax, p.value = pvalue))
}
```

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