Binomial Distribution

Binomial distribution

require('ggplot2')
x <- c(0:10)
y <- dbinom(x, size=10, .5, log = FALSE)
qplot(x, y, color = I("green"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

the Pascal triangle starts with: x = 0 size = 0 the density = 1 (the vertex of the triangle)

then we are moving to: x = c(0:1) size = 1 //one sample from the choise of two- 0 and 1 the density = 0.5, 0.5 … x = c(0:n) size = n probability = p

The thing that is freequently being forgotten is the way it behaves in asymmetric case, when the probability of a ‘positive’ outcome shifts towards 0 or 1.

y <- dbinom(x, size=10, .75, log = FALSE)
qplot(x, y, color = I("red"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

and

y <- dbinom(x, size=10, .25, log = FALSE)
qplot(x, y, color = I("magenta"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

The formula is

#dbinom formula (the function is in C)
dbin <- function(x, size, p) {
    d <- choose(size, x)*p^x*(1-p)^(size-x)
}
y <- dbin(x, size=10, .25)
qplot(x, y, size = I(2), color = I("magenta")) +geom_line(size =1, alpha =.5, color="red")