Random Number

Every random experiment results in two or more outcomes.
A variable whose values depend upon the outcomes of a random experiment is called a random variable denoted by capital letters X, Y, or Z and their values by the corresponding small letters x, y or z.

Random Numbers and their Generation

Random numbers are a sequence of digits from the set {0,1,2,⋯,9} so that, at each position in the sequence, each digit has the same probability 0.1 of being selected irrespective of the actual sequence, so far constructed.

The simplest ways of achieving such numbers are games of chance such as dice, coins, cards or by repeatedly drawing numbered slips out of a jar. These are usually grouped purely for convenience of reading but this would becomes very tedious for long runs of each digit. Fortunately tables of random digits are widely available now.

Pseudo Random Process

Pseudo Random Process is a process that appears to be random but actually it is not. Pseudo random sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process. Such a process is easier to produce than a genuinely random one, and has the benefit that it can be used again and again to produce exactly the same numbers and they are useful for testing and fixing software.

For implementation on computers to provide sequence of such digits easily, and quickly, the most common methods are called Pseudo Random Technique.

Here, digit will re-appear in the same order (cycle) eventually. For a good technique the cycle might be tens of thousands of digit long.
Of course the pseudo random digits are not truly random. In fact, they are completely deterministic but they do exhibit most of the properties of random digits. Generally, they methods involves the recursive formula e.g.

$X_{n+1}= a x_n +b\, mod\, m; n=0, 1, 2, …$

a, b and n are suitably chosen integer constants and the seed $x_0$ (a starting number i.e. n = 0) is an integer. (Note mode m means that if the result from formula is greater than m, then divide it by m and keep the remainder as a random number.

Use of this formula gives rise to a sequence of integers each of which is in the random 0 to m – 1.

Example

let a = 13, b=5, and m = 1000, Generate 500 random numbers.

Solution

$x_{n+1}=a x_n \,b\, mod\, 1000; n=0,1,2,…$

let seed $x_0=5$, then for n=0 we have

\begin{align*}
x_{0+1}&=13 \times 5 +5\, mod\, 1000=70\\
x_{1+1}&=13 \times 70 + 5\, mod\, 1000=915
\end{align*}

Application of Random Variables

The random numbers have wide applicability in the simulation techniques (also called Monte Carlo Methods) which have been applied to many problems in the various sciences and one useful in the situation where direct experimentation is not possible, the cost of conducting an experimetn is very high or the experiment takes too much time.

R code to Generate Random Number

# store the pseudo random output
rand.num<-numeric(500)
rand.seed<-5
for(i in 1:500){
rand.seed<-13*rand.seed+5
rand.num[i]<-rand.seed%%1000
}
rand.num

Pseudo Random Process

A pseudo random process is a process that appears to be random but actually it is not. Pseudo random sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process. Such a process is easier to produce than a genuinely random one, and has the benefit that it can be used again and again to produce exactly the same numbers and they are useful for testing and fixing software.

The generation of random numbers has many uses (mostly in Statistics, for Random Sampling, and Simulation, Computer Modeling, Markov Chains, Experimental Design). Before modern computing, researchers requiring random numbers would either generate them through various means like coin, dice, cards, roulette wheels, card shuffling etc or use existing random number tables.

A pseudo random variable is a variable which is created by a deterministic procedure (often a computer program or subroutine is used) which (generally) takes random bits as input. The pseudo random string will typically be longer than the original random string, but less random (less entropic, in the information theory sense). This can be useful for randomized algorithms.

Pseudo random numbers are computer generated random numbers and they are not truly random because there is an inherent pattern in any sequence of pseudo numbers.