Random number

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In statistics, a random number is a single observation (outcome) of a specified random variable. Where no distribution is specified, the continuous uniform distribution on the interval [0,1) is usually, but not always, intended.

In an informal sense, there is some circularity in this definition as the idea of random variable itself rests on the concept of randomness. A number itself cannot be random except in the sense of how it was generated.

Informally, selecting a number at random with uniform distribution on some set requires that all elements of that set were equally probable as outcomes before the selection. In particular, this means that knowledge of earlier numbers generated by this process, or some other process, do not yield any extra information about the next number. This is equivalent to statistical independence.

Statistical independence is insufficient in circumstances in which randomness, in the information theoretic sense of entropy is required. Consider the π, 3.14159..., which known to more than a trillion digits. They are statistically independent in that no pattern has ever been found in them, but they are useless for some purposes requiring randomness in the sense of entropy. After all, the digits are widely known and may be generated anew easily with computers, and so their entropy is 0. Any sequence with this little entropy can't used for some purposes such as the generation of key and nonces in cryptographic applications. The distinction is not merely academic, for insufficient entropy has often been the cause of security vulnerabilites in such systems.

Random number

See also