Canonical form parametrized normal distribution

CanonicalFormNormal.R

rcfnorm <- function(n, shape=0, inv_scale=1){
  rnorm(n, shape/inv_scale, 1/sqrt(inv_scale))
}

dcfnorm <- function(n, shape=0, inv_scale=1, ...){
  dnorm(n, shape/inv_scale, 1/sqrt(inv_scale), ...)
}

pcfnorm <- function(n, shape=0, inv_scale=1, ...){
  pnorm(n, shape/inv_scale, 1/sqrt(inv_scale), ...)
}

a1 = 1; b1 = 0.5
a2 = 2; b2 = 2.5

x1 = rcfnorm(10000, a1, b1)
x2 = rcfnorm(10000, a2, b2)
x = x1+x2
hist(x, breaks = "scott", freq=FALSE)

B = 1/(1/b1+1/b2)
A = B*(a1/b1+a2/b2)
curve(dcfnorm(x, A, B), add=TRUE, col="orangered")

plot(ecdf(x1+x2))
curve(pcfnorm(x, A, B), add=TRUE, col="orangered", lty=2)