Nothing
#' The Normal distribution
#'
#' @description
#' `r lifecycle::badge('stable')`
#'
#' The Normal distribution is ubiquitous in statistics, partially because
#' of the central limit theorem, which states that sums of i.i.d. random
#' variables eventually become Normal. Linear transformations of Normal
#' random variables result in new random variables that are also Normal. If
#' you are taking an intro stats course, you'll likely use the Normal
#' distribution for Z-tests and in simple linear regression. Under
#' regularity conditions, maximum likelihood estimators are
#' asymptotically Normal. The Normal distribution is also called the
#' gaussian distribution.
#'
#' @param mu,mean The mean (location parameter) of the distribution, which is also
#' the mean of the distribution. Can be any real number.
#' @param sigma,sd The standard deviation (scale parameter) of the distribution.
#' Can be any positive number. If you would like a Normal distribution with
#' **variance** \eqn{\sigma^2}, be sure to take the square root, as this is a
#' common source of errors.
#'
#' @details
#'
#' We recommend reading this documentation on
#' <https://pkg.mitchelloharawild.com/distributional/>, where the math
#' will render nicely.
#'
#' In the following, let \eqn{X} be a Normal random variable with mean
#' `mu` = \eqn{\mu} and standard deviation `sigma` = \eqn{\sigma}.
#'
#' **Support**: \eqn{R}, the set of all real numbers
#'
#' **Mean**: \eqn{\mu}
#'
#' **Variance**: \eqn{\sigma^2}
#'
#' **Probability density function (p.d.f)**:
#'
#' \deqn{
#' f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}
#' }{
#' f(x) = 1 / sqrt(2 \pi \sigma^2) exp(-(x - \mu)^2 / (2 \sigma^2))
#' }
#'
#' **Cumulative distribution function (c.d.f)**:
#'
#' The cumulative distribution function has the form
#'
#' \deqn{
#' F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx
#' }{
#' F(t) = integral_{-\infty}^t 1 / sqrt(2 \pi \sigma^2) exp(-(x - \mu)^2 / (2 \sigma^2)) dx
#' }
#'
#' but this integral does not have a closed form solution and must be
#' approximated numerically. The c.d.f. of a standard Normal is sometimes
#' called the "error function". The notation \eqn{\Phi(t)} also stands
#' for the c.d.f. of a standard Normal evaluated at \eqn{t}. Z-tables
#' list the value of \eqn{\Phi(t)} for various \eqn{t}.
#'
#' **Moment generating function (m.g.f)**:
#'
#' \deqn{
#' E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}
#' }{
#' E(e^(tX)) = e^(\mu t + \sigma^2 t^2 / 2)
#' }
#'
#' @seealso [stats::Normal]
#'
#' @examples
#' dist <- dist_normal(mu = 1:5, sigma = 3)
#'
#' dist
#' mean(dist)
#' variance(dist)
#' skewness(dist)
#' kurtosis(dist)
#'
#' generate(dist, 10)
#'
#' density(dist, 2)
#' density(dist, 2, log = TRUE)
#'
#' cdf(dist, 4)
#'
#' quantile(dist, 0.7)
#'
#' @export
dist_normal <- function(mu = 0, sigma = 1, mean = mu, sd = sigma){
mean <- vec_cast(mean, double())
sd <- vec_cast(sd, double())
if(any(sd[!is.na(sd)] < 0)){
abort("Standard deviation of a normal distribution must be non-negative")
}
new_dist(mu = mean, sigma = sd, class = "dist_normal")
}
#' @export
format.dist_normal <- function(x, digits = 2, ...){
sprintf(
"N(%s, %s)",
format(x[["mu"]], digits = digits, ...),
format(x[["sigma"]]^2, digits = digits, ...)
)
}
#' @export
density.dist_normal <- function(x, at, ...){
stats::dnorm(at, x[["mu"]], x[["sigma"]])
}
#' @export
log_density.dist_normal <- function(x, at, ...){
stats::dnorm(at, x[["mu"]], x[["sigma"]], log = TRUE)
}
#' @export
quantile.dist_normal <- function(x, p, ...){
stats::qnorm(p, x[["mu"]], x[["sigma"]])
}
#' @export
log_quantile.dist_normal <- function(x, p, ...){
stats::qnorm(p, x[["mu"]], x[["sigma"]], log.p = TRUE)
}
#' @export
cdf.dist_normal <- function(x, q, ...){
stats::pnorm(q, x[["mu"]], x[["sigma"]])
}
#' @export
log_cdf.dist_normal <- function(x, q, ...){
stats::pnorm(q, x[["mu"]], x[["sigma"]], log.p = TRUE)
}
#' @export
generate.dist_normal <- function(x, times, ...){
stats::rnorm(times, x[["mu"]], x[["sigma"]])
}
#' @export
mean.dist_normal <- function(x, ...){
x[["mu"]]
}
#' @export
covariance.dist_normal <- function(x, ...){
x[["sigma"]]^2
}
#' @export
skewness.dist_normal <- function(x, ...) 0
#' @export
kurtosis.dist_normal <- function(x, ...) 0
#' @export
Ops.dist_normal <- function(e1, e2){
ok <- switch(.Generic, `+` = , `-` = , `*` = , `/` = TRUE, FALSE)
if (!ok) {
return(NextMethod())
}
if(.Generic == "/" && inherits(e2, "dist_normal")){
abort(sprintf("Cannot divide by a normal distribution"))
}
if(.Generic %in% c("-", "+") && missing(e2)){
e2 <- e1
e1 <- if(.Generic == "+") 1 else -1
.Generic <- "*"
}
if(.Generic == "-"){
.Generic <- "+"
e2 <- -e2
}
else if(.Generic == "/"){
.Generic <- "*"
e2 <- 1/e2
}
# Ops between two normals
if(inherits(e1, "dist_normal") && inherits(e2, "dist_normal")){
if(.Generic == "*"){
abort(sprintf("Multiplying two normal distributions is not supported."))
}
e1$mu <- e1$mu + e2$mu
e1$sigma <- sqrt(e1$sigma^2 + e2$sigma^2)
return(e1)
}
# Ops between a normal and scalar
if(inherits(e1, "dist_normal")){
dist <- e1
scalar <- e2
} else {
dist <- e2
scalar <- e1
}
if(!is.numeric(scalar)){
abort(sprintf("Cannot %s a `%s` with a normal distribution",
switch(.Generic, `+` = "add", `-` = "subtract", `*` = "multiply", `/` = "divide"),
class(scalar)))
}
if(.Generic == "+"){
dist$mu <- dist$mu + scalar
}
else if(.Generic == "*"){
dist$mu <- dist$mu * scalar
dist$sigma <- dist$sigma * abs(scalar)
}
dist
}
#' @method Math dist_normal
#' @export
Math.dist_normal <- function(x, ...) {
# Shortcut to get log-normal distribution from Normal.
if(.Generic == "exp") return(vec_data(dist_lognormal(x[["mu"]], x[["sigma"]]))[[1]])
NextMethod()
}
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.