Probability Tutorials - Random Variables
The concept of Random Variable is a natural extension of the concept
of a random experiment. Let's recall for that a random
experiment is simply a procedure that leads to a non-deterministic
outcome (meaning, we cannot predict it beforehand).
For example, a random experiment corresponds to toss a coin. You
cannot predict the outcome (can you?), and no matter how much you
practice you won't be able to obtain head or tail at will. Another
example, say that you cast a die. If the die is reasonably fair, you
won't be able to predict the number you get every time you cast the
die (go and tell those guys in Vegas...)
-- Now, a random variable  corresponds to a a function that
assigns a number to the outcomes of a random experiment.
Huh?? (That's what you ask....)
Ok, bare with me for a second. Come back to the random experiments.
Say you cast two dice, to make it more exciting. What are the
possible outcomes of you experiments? Well, it is going to be all
the possible pairs ) , with  . (Or you
can write them the long way (1, 1), (1, 2), (1, 3),....(6, 6)). So,
a random variable would be, for instance, the sum of the numbers
shown on the dice.
For example, if the outcome is (1, 2), then the random variable
corresponds to the sum of the numbers, which is  . You
see, is in fact a random variable, because it assigns a
number to the outcomes of a random experiment. Why we call it random
variable? Because it is random too! You cannot predict the value of
a random variable before hand. Once you have the outcome of the
random experiment, just then you know the value of the random
variable.
Now we give the technical definition of a random variable, even
though the concepts described above are enough to proceed, and keep
learning more about random variables.
Definition: Let  be the sample space of
the random experiment  . We say that is a random
variable when is a function from to  :
This definition is saying exactly the same we said before. Some
other examples of random variables:
Example: Assume that you toss a fair coin 3 times. We
define the random variable as the total number of heads. Another
random variable  is defined as the total number of tails.
(...to be continued)
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