Properties of Random Numbers

It’s very important to have random numbers but achieving true randomness on purpose is practically impossible.

#Statistical Randomness

A random number is a result of a random variable combination specified by a probability distribution function. When no distribution is specified, it is assumed that the continuous uniform distribution in the interval [0, 1] is used.

A numerical sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness. Statistical randomness does not necessarily imply “true” randomness, i.e., objective unpredictability. Pseudo-randomness is sufficient for many uses, such as statistics, hence the name statistical randomness.

A sequence of random numbers must have 2 important properties

Uniformity: All numbers have the same probability of appearing.
Independence: The current value of a random variable has no relation to previous values.

The first step to simulate numbers from a distribution is to be able to independently simulate random numbers from a continuous uniform distribution between 0 and 1.

#Uniform Distribution [0,1]

To extract a random value x that represents a number between 0 and 1, given that the random variable x takes any value of this interval with all values being equally likely, it is said that the random variable x has a continuous uniform probability distribution.

PDF

$$ f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases} $$ where a=0 and b=1, so $$ f(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} $$

With the cumulative distribution function, we can determine the probability that a random number we take from this interval is less than or equal to a certain value.

CDF

$$ F(x) = \begin{cases} 0, & \text{for } x < a, \\ \frac{x-a}{b-a}, & \text{for } a \leq x \leq b, \\ 1, & \text{for } x > b. \end{cases} $$ where a=0 and b=1, so $$ F(x) = \begin{cases} 0, & x < 0, \\ x, & 0 \leq x \leq 1, \\ 1, & \text{otherwise}. \end{cases} $$

Mean = 1/2 (a+b). Variance = 1/12 $(b-a)^2$

#Uniformity

Testing Uniformity Example of uniform vs non-uniform:

#Independence

The second requirement the random numbers of the sequence generated need to respect is independence. This means that the probability of observing a value in a particular sub-interval of (0,1) is independent of the previous values drawn. Testing Independence