Probability and Statistics

Probability Histogram

Here we write a simple function to produce a graph for "probability mass function" on the top of histogram. This special function prob.mass() is written as follows:

prob.mass = function(x,prob,...){
  for(i in 1:length(x)){
    polygon(c(x[i]-0.5,x[i]-0.5,x[i]+0.5,x[i]+0.5),
            c(0,prob[i],prob[i],0),...)
  }
}

Programming Note. If the argument “...” is supplied, partial arguments intended for “...” will be swallowed and matched as “...” The function polygon(x,y,...) draws the polygon surrounded by $(x_1,y_1),\ldots,(x_n,y_n),(x_1,y_1)$ .

Random sample. When we obtain the outcomes of random variable from the repeated experiment, the collection of outcomes is called a "random sample." To generate a random sample of size n from a probability mass function prob, we use

data = sample(x,size=n,replace=T,prob=prob)

The first argument x is either a positive integer or an array of values $a_1,a_2,\ldots,a_k$ corresponding to the probability mass function $p(a_i)$

. If x is an integer, the sample takes values from 1 to x.

prob = c(1/10,2/10,3/10,4/10)
x = length(prob)
n = 500
data = sample(x,size=n,replace=T,prob=prob)
hist(data, freq=F, breaks=seq(min(x)-0.5,max(x)+0.5,by=1.0), ylim=c(0,max(prob)+0.1), col='green')
prob.mass(x,prob,lwd=2,border='red')
print(data)

Here we generates the sample x of size n representing the number of outcomes from 1 to 4 according probability prob.

Sample R code. You can download histdemo.R, and run it.

Programming Note. We can introduce a structure of the numeric vector by

vec = c(10.4, 5.6, 3.1, 6.4, 21.7)

It can take an arbitrary number of values, and set the length for the variable vec.