A random number generator (RNG) is a device that generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. Random number generators can be true hardware random-number generators (HRNG), which generate genuinely random numbers, or pseudo-random number generators (PRNG) which generate numbers which look random, but are actually deterministic, and can be reproduced if the state of the PRNG is known.
So a pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG’s seed (which may include truly random values).
There exist several computational methods for pseudo-random number generation, but all fall short of the goal of true randomness, although they may meet, with varying success, some of the statistical tests for randomness intended to measure how unpredictable their results are.
The generation of pseudo-random numbers is an important and common task in computer programming.
There are a couple of methods to generate a random number based on a probability density function. These methods involve transforming a uniform random number in some way. Because of this, these methods work equally well in generating both pseudo-random and truly random numbers.
One method, called the inversion method, involves integrating up to an area greater than or equal to the random number (which should be generated between 0 and 1 for proper distributions).
A second method, called the acceptance-rejection method, involves choosing an x and y value and testing whether the function of x is greater than y value. If it is, the x value is accepted. Otherwise, the x value is rejected and the algorithm tries again.
Random numbers uniformly distributed between 0 and 1 can be used to generate random numbers of any desired distribution by passing them through the inverse cumulative distribution function (CDF) of the desired distribution (see Inverse transform sampling). Inverse CDFs are also called quantile functions.
A PRNG suitable for cryptographic applications is called a cryptographically secure PRNG (CSPRNG). A requirement for a CSPRNG is that an adversary not knowing the seed has the only negligible advantage in distinguishing the generator’s output sequence from a random sequence. In other words, while a PRNG is only required to pass certain statistical tests, a CSPRNG must pass all statistical tests that are restricted to polynomial time in the size of the seed.
At this link is it possible to find the list of the most common PRGs.