What is the difference between a multivariate normal random deviate and a random variable? More specifically, I suppose, what exactly is a deviate?
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My understanding is that a deviate/variate a particular outcome of random variable. If I generate a 100 draws from standard normal (a random variable) using my favorite PRNG, I get 100 standard normal deviates.
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$\begingroup$ I've always regarded normal deviates as an oxymoron that has been perpetrated by statisticians as an inside joke.After all, in pop psychology, some peop[e are normal, and any one who is not normal is a deviate! More seriously, to me a deviate suggests a sample value of a zero-mean random variable; the numerical value is the deviation from the mean. But then, what do I know? I am neither a statistician nor a psychologist, or for that matter, a linguist or lexicographer. $\endgroup$ Commented Aug 3, 2018 at 13:46
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The terms "variate" and "deviate" appear to be interchangeable:
a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on. In that context, and in statistics, those values are known as a random variates, or occasionally random deviates[1]
[1] https://en.wikipedia.org/wiki/Random_variate (3/8/2018)
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$\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$– FerdiCommented Aug 3, 2018 at 11:45
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$\begingroup$ More tersely, a random variate is an element of the image of a random variable. $\endgroup$– GalenCommented Aug 15, 2023 at 3:27
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A variable (singular) can and usually does vary. A variate (singular) never varies - it is a realisation, a raw value. A deviate is the output of a calculation, not a raw datum. As the name suggests, every variate becomes a deviate once subtracted from the mean, for example. The deviate-du-jour divided by the average deviate is the T statistic. ie The SNR.
random variableand removed the tagsvarianceandstandard-deviationwhich don't directly relate to the question. A random variable might have a multivariate normal distribution, or it might (and typically will) have some other distribution. 'Deviate', as Dimitry suggests, tends mostly to be reserved for an outcome of a random variable. $\endgroup$