How to Generate Pseudorandom Numbers | Infinite Series

Summary

This video explores the concept of random number generation in computers, highlighting the distinction between truly random and pseudo-random numbers. It covers physical random number generators, like Geiger counters, and pseudo-random methods such as the middle-square algorithm and linear congruential generators. The video also discusses statistical tests for randomness, the significance of seed values in replicating sequences, and how pseudo-random numbers can simulate different distributions, such as those representing human height. The video concludes by demonstrating how deterministic processes can create sequences with properties resembling true randomness, applicable in fields like encryption and simulations.

Takeaways

• 😀 Computers need random numbers for encryption, simulations, and games, but they are deterministic and cannot naturally generate true randomness.
• 😀 True random numbers can be generated using physical phenomena, like a Geiger counter measuring background radiation, but this requires extra hardware and is slow.
• 😀 Pseudo-random numbers mimic randomness and are generated using deterministic processes, meaning they are not truly random but appear random in practice.
• 😀 The middle-square algorithm, proposed by John von Neumann, generates random-looking numbers by squaring a seed number and extracting the middle digits.
• 😀 The linear congruential generator (LCG) is a more efficient and widely-used method for generating pseudo-random numbers using a simple formula based on a seed value.
• 😀 Pseudo-random number generators (PRNGs) are reproducible, meaning they can generate the same sequence of numbers from the same seed, which is useful for testing programs.
• 😀 The period of a pseudo-random number sequence refers to how many terms can be generated before the sequence starts repeating.
• 😀 The middle-square algorithm has a very short period, often cycling in just a few steps, making it less reliable for generating large sequences.
• 😀 In contrast, the linear congruential generator can have a much longer period, especially when the modulus, multiplier, and increment values are carefully chosen.
• 😀 To generate numbers that follow specific distributions (like normal distribution), we can use inverse transform sampling, which transforms uniformly distributed numbers into numbers with the desired distribution.

Q & A

How can a deterministic computer produce genuinely random numbers?

A deterministic computer can produce genuinely random numbers by relying on external physical phenomena, such as measuring background radiation with hardware like a Geiger counter. These sources are assumed to be random, but they have limitations like the need for extra hardware and slower speed.

What is the difference between random and pseudo-random numbers?

Random numbers are generated from unpredictable physical processes, whereas pseudo-random numbers are generated by deterministic algorithms, meaning they follow a specific, repeatable pattern but can still appear random statistically.

Why do computers often use pseudo-random number generators instead of true random number generators?

Pseudo-random number generators are often preferred because they are computationally efficient, do not require extra hardware, and produce reproducible sequences when given the same seed. True random number generators, on the other hand, are slower and not reproducible.

How does the middle-square algorithm generate pseudo-random numbers?

The middle-square algorithm generates pseudo-random numbers by squaring a given seed number, extracting the middle digits of the result, and using them as the next number in the sequence. This process is repeated iteratively to produce a sequence of random-looking numbers.

What is the linear congruential generator and how does it work?

The linear congruential generator (LCG) is a type of pseudo-random number generator that uses a formula involving a modulus, multiplier, increment, and seed. The formula produces a sequence of numbers by repeatedly applying the rule: (a * x_n + c) mod m, where 'a' is the multiplier, 'c' is the increment, 'm' is the modulus, and 'x_n' is the previous number in the sequence.

What is the period of a pseudo-random number sequence?

The period of a pseudo-random number sequence refers to the number of terms generated before the sequence repeats itself. For example, the middle-square algorithm has a small period (often smaller than 4,096), while a well-chosen linear congruential generator can have a much longer period.

What are the advantages of pseudo-random number generators over true random number generators?

Pseudo-random number generators are advantageous because they are faster, computationally efficient, and reproducible with the same seed. True random number generators, in contrast, require extra hardware, are slower, and cannot reproduce the same sequence of random numbers.

Why is it important for pseudo-random numbers to appear random, and how is randomness measured?

Pseudo-random numbers are used in applications like encryption and simulations, where they need to mimic randomness. Randomness can be measured statistically using tests like calculating the running average, plotting histograms, and checking for recurring patterns in the sequence.

How can you generate numbers with specific distributions, such as a normal distribution, using pseudo-random numbers?

To generate numbers with specific distributions, like the normal distribution, you can use inverse transform sampling. By starting with a uniform distribution of numbers between 0 and 1, you can map these numbers to the desired distribution by finding corresponding values on the cumulative distribution function (CDF).

What is the cumulative distribution function (CDF), and how does it help in generating numbers from a specific distribution?

The cumulative distribution function (CDF) shows the probability that a random variable will take a value less than or equal to a certain point. In the context of generating specific distributions, you can use the CDF to map uniform random numbers to the desired distribution by finding the corresponding value for each random number.