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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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Thanks, I have already understood that. Are there other usages of variates in mathematics? –  Tim Sep 30 '12 at 3:29
    
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. –  vol7ron Jul 11 '13 at 17:51
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