random variable
In plain wordsA quantity whose value depends on chance, such as the number rolled on a die or the next word a chatbot will write.
A function that assigns a number (or vector) to each outcome of a random experiment, so that probabilities of outcomes become probabilities of values; formally, a measurable function on a probability space. Random variables are discrete (counts, class labels, tokentoken, n.: The small chunk of text an AI language model reads and writes — often a word, part of a word or a punctuation mark. AI usage, limits and prices are counted in tokens. indices) or continuous (heights, the weights of a network at initialization), and each has a probability distributionprobability distribution, n.: A complete description of how likely each possible outcome is — for a fair die, one-sixth for each face. A language model produces one of these over its whole vocabulary at every step. summarized by quantities such as its expected valueexpected value, n.: The long-run average you would get if a chancy situation could be repeated many times — 3.5 for a fair die, even though no single roll shows 3.5. and variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves.. Machine learning treats data points, labels, mini-batch gradients, sampled tokens and rewards — and, in Bayesian inferenceBayesian inference, n.: Reasoning that treats unknowns as uncertain beliefs and updates them with evidence using Bayes' theorem. The result is not one answer but a range of answers, each with a probability., model parameters — as random variables; by convention they are written in capitals (X) and their observed values in lower case (x). Independence, correlationcorrelation, n.: A measure of how closely two quantities move together, from −1 (one rises as the other falls) through 0 (no straight-line link) to +1 (they rise together). It does not show that one causes the other. and mutual informationmutual information, n.: How much knowing one thing tells you about another — for example, how much a word reveals about the topic of a document. It is zero when the two are unrelated. describe how random variables relate to one another.
Treat the reward as a random variable and report its mean and variance over 32 rollouts.
Origin
The idea is implicit in early probability, but the name is 20th-century: 'random variable' is recorded in Biometrika in 1914, Francesco Paolo Cantelli wrote variabile casuale in 1916, Andrey Kolmogorov formalized the notion in 1933, and Harald Cramér's 1937 monograph Random Variables and Probability Distributions spread the term.
Related
- probability distributionprobability distribution, n.: A complete description of how likely each possible outcome is — for a fair die, one-sixth for each face. A language model produces one of these over its whole vocabulary at every step.
- expected valueexpected value, n.: The long-run average you would get if a chancy situation could be repeated many times — 3.5 for a fair die, even though no single roll shows 3.5.
- variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves.
- i.i.d.i.i.d., abbr.: Short for 'independent and identically distributed': every data point is produced the same way, and none influences any other — like repeated rolls of one fair die.
- stochasticstochastic, adj.: Involving chance, so the same starting point can lead to different outcomes. AI chatbots are stochastic: ask the same question twice and you may get two different answers.
- probabilityprobability, n.: How likely something is, on a scale from 0 for the impossible to 1 for the certain; rolling a six with a fair die scores one-sixth. AI systems state their predictions, even each next word, as probabilities.
Mentioned in
probabilityprobability, n.: How likely something is, on a scale from 0 for the impossible to 1 for the certain; rolling a six with a fair die scores one-sixth. AI systems state their predictions, even each next word, as probabilities., entropyentropy, n.: A measure of unpredictability. A fair coin toss has more entropy than a rigged one; an AI that is unsure of its next word has high entropy., expected valueexpected value, n.: The long-run average you would get if a chancy situation could be repeated many times — 3.5 for a fair die, even though no single roll shows 3.5., i.i.d.i.i.d., abbr.: Short for 'independent and identically distributed': every data point is produced the same way, and none influences any other — like repeated rolls of one fair die., law of large numberslaw of large numbers, n.: The mathematical rule that averages of chance events steady out over many repetitions: flip a fair coin a few times and the share of heads can be far from one-half, but over thousands of flips it gets very close., Markov chainMarkov chain, n.: A chance-driven process that moves between states without memory: the odds of its next move are set by where it is, never by the path that led there. Forecasting tomorrow's weather from today's alone is a simple example., probability distributionprobability distribution, n.: A complete description of how likely each possible outcome is — for a fair die, one-sixth for each face. A language model produces one of these over its whole vocabulary at every step., stochasticstochastic, adj.: Involving chance, so the same starting point can lead to different outcomes. AI chatbots are stochastic: ask the same question twice and you may get two different answers., variancevariance, n.: A number that grows the more widely something's possible values scatter away from its average, and is zero if it never varies. Its square root, the standard deviation, uses the same units as the values themselves., covariancecovariance, n.: A measure of whether two quantities tend to rise and fall together, move in opposite directions, or show no consistent relationship., Jensen's inequalityJensen's inequality, n.: A rule saying that for a bowl-shaped curve, the average of the outputs is at least the output of the average. It is the hidden step behind many proofs in machine learning., mutual informationmutual information, n.: How much knowing one thing tells you about another — for example, how much a word reveals about the topic of a document. It is zero when the two are unrelated., probability density functionprobability density function, n.: A curve showing where a measurement is likely to fall. The area under the curve between two values gives the chance of landing between them.