Properties of Random Numbers

• A sequence of random numbers, $R_1, R_2, R_3, ...$ must have two important properties:
  1. uniformity, i.e. they are equally probable every where
  2. independence, i.e. the current value of a random variable has no relation with the previous values

• Each random number $R_i$ is an independent sample drawn from a continueous uniform distribution between zero and one.
  • pdf
    
    $$f(x) = \left\{ \begin{array}{ll} 1 & 0 \le x \le 1 \\ 0 & {\rm otherwise} \end{array} \right.$$
    
    

  • expectation
    
    $$E(R) = \int_0^1 x dx = \frac{1}{2}$$
    
    

  • variance
    
    $$V(R) = \int_0^1 x^2 dx - [E(R)]^2 = \frac{1}{12}$$
    
    

• Some consequences of the uniformity and independence properties
  1. If the interval (0,1) is divided into n sub-intervals of equal length, the expected number of observations in each interval is N/n where N is the total number of observations. Note that N has to be sufficiently large to show this trend.
  2. The probability of observing a value in a particular interval is independent of the previous values drawn.