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Random Variable

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Dan Dolderman

Measurement as a random variable: Let’s agree that • a measurement is an activity, which depends on certain, known or unknown conditions • the outcome of each measurement is understandably varying due to the changing parameters • measurement is at the heart of a scientific experiment • An essential characteristic of a scientific experiment is its reproducibility. Due to the variability of the outcome of each measurement, of the same phenomenon under the same conditions, we consider a measurement as a ‘random variable’. A random variable is a process, the outcome of which is a random selection of a number from a collection of numbers (known as the range of the random variable.) Of course, a random variable is selecting numbers based on absolutely no algorithm, and this is what characterizes the random variable. As soon as any element of uncertainty involves in a decision making, random variables are used as mathematical models. We usually build a non-deterministic model for our knowledge of the reality, and at the center of this model there resides a random variable. But often we face a certain decision making that is only partially random. The existing information about the nature of the selection/experiment usually influences the outcome of the selection; as such this randomness is no longer absolutely random, yet it is random as it does not obey an algorithm for its decision making. Thus, in various investigations we use one of the many types of random variables that may better suit the nature of our selection. The outcome of a random variable is a number, but what is it that this number is supposed to be representing? Is there an absolute measurement that this random variable is trying to capture, or are these random values collectively presenting us with a kind of information about what we are trying to measure? Or perhaps these random values are collectively trying to invent the measurement for us! In other words, is there one actual number that we do not know (but we believe it exists) that we are trying to achieve, or is it that there is no actual number and we are just 1 constructing it by trying to measure it ? Here is an example to facilitate this thought: 1 This is an example of the mystic gaze, ancient among the mystics, and wathextensively presented in a book of verses called the ‘conference of the birds’ by the 12 century mystic Attar of Nishapur. In this epic, the birds are searching their leader, the legendary Phoenix. In this search many of them perish, but only 30 of them have enough endurance and enough faith in the existence of their leader (the absolute truth), and only this group finishes the using the projector we project the picture of a dart board on the wall and ask people to try to throw dart at the imaginary/projected dart board. Of course many try it and some come very close to hitting the center and some even manage to hit the center. Of course at this point we are sure that only few of the darts landed on the center, and in this sense the rest of the darts landed off the target, and as such they are ‘incorrect’. Then we turn the projector light off, and we look at the dart holes on the wall. At this point it is only the holes that lead us to believe that there was a center and that the dart holes were many attempts to hit the center. But it is not clear which dart holes belong to the ones that hit the target: which ones are the ‘correct’ ones and which ones are the ‘incorrect’ ones. But still collectively the dart holes will tell us a story as to where the center could be; and to be sure we really need all of the dart holes to complete the picture. In a sense the truth belongs to the entire population of the dart holes: together they tell a story; it is together that the population talks about being close to the center or being far from it. Each dart hole, even the furthest away from the center is having a share in indicating where the center could be, and as such each dart hole carries a portion of the truth. Now without any knowledge of the center we try to use the dart holes to actually recreate the center. In practice we do not insist on the knowledge of the center but we use the probabilities of the dart holes being close to it or far from it: this is the language of the modern sciences; closer to political elections when they report the results a poll they generally mention a number with an error together with a probability (for example the conservatives are gaining 40% of the votes, with the error of 2%, 18 time out of 20.) This is how a measurement is to be discussed in the modern sciences. Let’s go back to the example of the imaginary dart board and that the furthest point carries some truth: if we eliminate the furthest point then the chances of other points being more true is increasing, and as such the distribution will become distorted (and away from the true reality of itself.) here is somewhat related paradox: three prisoners A,B, and C, are in a cell when the guard announces that two of the prisoners will be released and the third one will be sentenced harshly, but the guard is not willing to reveal the identity of the prisoner who will be harshly sentenced. As such each prisoner will have 1/3 chances of being sentenced harshly and the 2/3 chances of going free. At this time prisoner A approaches the guard and asks the guard: “ we both know that surely one of the two prisoners B or C will be set free; will you be kind to tell me which one of the prisoners B or C will be set free?” The guard suggests that if he declares one of the prisoners B or C as one who will surely go home, then the chances of A being harshly sentenced will increase from 1/3 to ½ and that is not fair to A. As you can see the elimination of one of the options (who was surely going to go home) will alter the nature of the distribution. Any experiment seems to suggest one type of randomness inherent in it. For example the experiment of flipping a coin can have only two outcomes: {Head,Tail} or if you journey to the mount Olympus, where they hear a disembodied voice which welcomes them and calls them the various attributes of the Phoenix: they had become the truth (which they searched so faithfully!) wish to represent them numerically, the domain of the associated random variable is {0,1}. Here is a classification of various types of random variables suitable for various experiments: •Bernoulli random variable has range of values {0,1}; otherwise known as Success and Failure, or True and False. •Uniform random variable has a range of values {1,2,3,…,n} for some number n. For example throwing a fair die presents us with 6 outcomes each of which is equally likely. This can help modeling the experiments that have one of the n different outcomes all are equally likely because we have absolutely no specific knowledge about the details of the experiment. •Binomial random variable has a range of values {0,1,2,3, …, n} for some number n, and this is used to model the number of successes in n trials of a Bernoulli experiment. For example we flip a coin n times and count the number of the Heads; this number can be modeled by a binomial random variable. •Geometric random variable has infinite range {1,2, 3, … } of values and can be most naturally used to model an experiment that involves an attempt to get one success only. We try the Bernoul
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