Random Numbers From Outer Space

In a rare but much appreciated break from the Nixie tube norm of clock making, [Alpha-Phoenix] has designed a muon-powered random number generator around that warm, vintage glow. Muons are subatomic particles that are like electrons, but much heavier, and are created when pions enter the atmosphere and undergo radioactive decay. The Geiger-Müller tube, mainstay of Geiger counters the world over, detects these incoming muons and uses them to generate the number.

Inside the box, a 555 in astable mode drives a decade counter, which outputs the numbers 0-9 sequentially on the Nixie via beefy transistors. While the G-M tube waits for muons, the numbers just cycle through repeatedly, looking pretty. When a muon hits the tube, a second 555 tells the decade counter to stop immediately. Bingo, you have your random number! The only trouble we can see with this method is that if you need a number right away, you might have to go get a banana and wave it near the G-M tube.

Usually if you want to detect radiation from space you would have a vertical array of GM tubes, the idea being that if all the tubes are set off, then you know the radiation came from directly above or below, and below can be shielded with a chunk of lead. If only one or two tubes get set off, then the radiation came from the side and is likely terrestrial in origin.

"Random" means just that: it's random. Every value in the range has the same probability of being chosen, regardless of what's been chosen before. So even if it picked the number 5, for instance, it still has the same chance of picking 5 again as it does of picking any other number. You shouldn't expect random numbers to avoid duplicates -- if they did, they wouldn't be random :)

Since you are generating a relatively small number of random numbers from a large sample you should be able to regenerate a new number on collision. Adding random nums until you get to 500 will result in a few extra calls to the random generator, but it will guarantee 500 unique numbers:

Unilateral peripheral vestibular deficit leads to broad cognitive difficulties and biases in spatial orientation. More specifically, vestibular patients typically show a spatial bias toward their affected ear in the subjective visual vertical, head and trunk orientation, fall tendency, and walking trajectory. By means of a random number generation task, we set out to investigate how an acute peripheral vestibular deficit affects the mental representation of numbers in space. Furthermore, the random number generation task allowed us to test if patients with peripheral vestibular deficit show evidence of impaired executive functions while keeping the head straight and while performing active head turns. Previous research using galvanic vestibular stimulation in healthy people has shown no effects on number space, but revealed increased redundancy of the generated numbers. Other studies reported a spatial bias in number representation during active and passive head turns. In this experiment, we tested 43 patients with acute vestibular neuritis (18 patients with left-sided and 25 with right-sided vestibular deficit) and 28 age-matched healthy controls. We found no bias in number space in patients with peripheral vestibular deficit but showed increased redundancy in patients during active head turns. Patients showed worse performance in generating sequences of random numbers, which indicates a deficit in the updating component of executive functions. We argue that RNG is a promising candidate for a time- and cost-effective assessment of executive functions in patients suffering from a peripheral vestibular deficit.

In the formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space. This allows consideration of the pushforward measure, which is called the distribution of the random variable; the distribution is thus a probability measure on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be independent.

It is common to consider the special cases of discrete random variables and absolutely continuous random variables, corresponding to whether a random variable is valued in a discrete set (such as a finite set) or in an interval of real numbers. There are other important possibilities, especially in the theory of stochastic processes, wherein it is natural to consider random sequences or random functions. Sometimes a random variable is taken to be automatically valued in the real numbers, with more general random quantities instead being called random elements.

The term "random variable" in statistics is traditionally limited to the real-valued case ( E = R {\displaystyle E=\mathbb {R} } ). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.

However, the definition above is valid for any measurable space E {\displaystyle E} of values. Thus one can consider random elements of other sets E {\displaystyle E} , such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type E {\displaystyle E} , or an E {\displaystyle E} -valued random variable.

This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of E {\displaystyle E} , using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space Ω {\displaystyle \Omega } , which allows the different random variables to covary). For example:

Recording all these probabilities of outputs of a random variable X {\displaystyle X} yields the probability distribution of X {\displaystyle X} . The probability distribution "forgets" about the particular probability space used to define X {\displaystyle X} and only records the probabilities of various output values of X {\displaystyle X} . Such a probability distribution, if X {\displaystyle X} is real-valued, can always be captured by its cumulative distribution function

In examples such as these, the sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.

The possible outcomes for one coin toss can be described by the sample space Ω = { heads , tails } {\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}} . We can introduce a real-valued random variable Y {\displaystyle Y} that models a $1 payoff for a successful bet on heads as follows: Y ( ω ) = { 1 , if ω = heads , 0 , if ω = tails . {\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}

In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of F {\displaystyle {\mathcal {F}}} is a measurable subset of possible outcomes, the function P {\displaystyle P} gives the probability of each such measurable subset, E {\displaystyle E} represents the set of values that the random variable can take (such as the set of real numbers), and a member of E {\displaystyle {\mathcal {E}}} is a "well-behaved" (measurable) subset of E {\displaystyle E} (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.

When E {\displaystyle E} is a topological space, then the most common choice for the σ-algebra E {\displaystyle {\mathcal {E}}} is the Borel σ-algebra B ( E ) {\displaystyle {\mathcal {B}}(E)} , which is the σ-algebra generated by the collection of all open sets in E {\displaystyle E} . In such case the ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is called an E {\displaystyle E} -valued random variable. Moreover, when the space E {\displaystyle E} is the real line R {\displaystyle \mathbb {R} } , then such a real-valued random variable is called simply a random variable.

If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted X = d Y {\displaystyle X{\stackrel {d}{=}}Y} ) if they have the same distribution functions:

To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform. 2b1af7f3a8