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I have two questions:

1. Which one is determined first: (i) Random variable's values, (ii) the random experiment?

For instance this link says the following:

Suppose a variable X can take the values 1, 2, 3, or 4.

The probabilities associated with each outcome are described by the following table:

Outcome     1   2   3   4
Probability 0.1 0.3 0.4 0.2

The probability that X is equal to 2 or 3 is the sum of the two probabilities: P(X = 2 or X = 3) = P(X = 2) + P(X = 3) = 0.3 + 0.4 = 0.7. Similarly, the probability that X is greater than 1 is equal to 1 - P(X = 1) = 1 - 0.1 = 0.9, by the complement rule.

Which means, the values of Random Variables are decided first which is very unusual.

2. Can a Random Variable take random values?

For instance, can we roll a die and a random variable associated with it take on,

X = {99, 0, 100, 1, -35, 2.4} ?

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    $\begingroup$ The random experiment defines the random variable if you associate the values of the outcomes and probabilities with the random variable You can define a random variable as you did for X in part 2 by say associating the die outcome 1 with 99, 2 with 0, 3 with 100, 4 with 1, 5 with .-35 and 6 with 2.4. Presumably you would assign 1/6 as the probability for each of the 6 possible outcomes. $\endgroup$
    – Michael R. Chernick
    Apr 5, 2018 at 5:10
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    $\begingroup$ I am not sure what do you mean "the values of Random Variables are decided first which is very unusual". The Random variable is a measurable function, i.e probabilities of events involving the function can be computed, it is a function on the events (certain sets) . $\endgroup$
    – Deep North
    Apr 5, 2018 at 5:10

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