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From Wikipedia

A random variate is a particular outcome of a random variable.

If I understand correctly, a random variable is a measurable mapping, and a random variate is just a member of the codomain of a random variable.

In general, what differences are between variable and variate in mathematics? What do they mean respectively?

Thanks!

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A random variable or stochastic variable is a variable whose value is subject to variations due to chance (from Wiki).

A random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable would have different values (also from Wiki).

Suppose $X$ is a random variable which stands for the outcome of tossing a fair dice. So $X$ can take value from $1$ through $6$ with equal probability of $1/6$. Now you actually toss a dice and get a number $4$. This number is a particular outcome of $X$, and thus a random variate. If you toss again, you may get another different value.

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  • $\begingroup$ Thanks, I have already understood that. Are there other usages of variates in mathematics? $\endgroup$ – StackExchange for All Sep 30 '12 at 3:29
  • $\begingroup$ Somewhat old question/answer, and I'm somewhat confused on the outcome of a random variable as outcome makes me think there is another function acting on the variable; but to clarify, my understanding of the above is that a variate is a variable's value at a particular point/instance. $\endgroup$ – vol7ron Jul 11 '13 at 17:51
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Some additional remarks (both from Wikipedia page):

The distinction between random variable and random variate is subtle and is not always made in the literature. It is useful when one wants to distinguish between a random variable itself with an associated probability distribution on the one hand, and random draws from that probability distribution on the other, in particular when those draws are ultimately derived by floating-point arithmetic from a pseudo-random sequence.

and:

In probability theory, 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, and this represents a wider meaning than just that associated with pseudorandom numbers.

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I have a feeling that the term Variate goes back to good old R A Fisher, were the more sophisticated definition of a Random Variable had not yet developed. About twenty years ago many text books defined the term variate as the actual numeric outcome of an actual random experiment. For example the weight of a mouse was called a variate. If I measured the weights of a number of mice, these values would be called variate's. This is quite confusing because it seem to be confusing the value of a random variable with the variable itself. One could say two recorded random values is "Bi variate" ?? What seems to have happened is that the meaning has changed and that virtually the term Random variable and Variate virtually mean the same thing. So that when I say Bi-variate data I mean data obtained form two random variables defined over the same sample space. I have no idea if what I have said has become formalised but looking over old texts has seen that a gradual change in definition has occurred. The trouble is that many teachers just dismiss the term Random variable (Variate) as if it were an algebraic variable when in fact they are quite different ideas.

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Here are the definitions from the Concise Oxford English Dictionary (11th ed.):

Variate: "a quantity having a numerical value for each member of a group, especially one whose values occur according to a frequency distribution."

Variable: "a factor or quantity able to assume different numerical values."

The Oxford English Dictionary has more or less identical definitions, and various quotes from references in statistics and mathematics to back them up.

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