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Random Variables *May 26, 2007*

*Posted by Peter in Exam 1/P.*

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Suppose you have a standard, fair, 6-sided die. We are interested in the possible outcomes of rolling this die. Naturally, the outcomes of multiple rolls of the die are not predetermined or fixed, but rather are the result of a *random* process. And yet, “random” doesn’t mean that we don’t have any information about what the possible outcomes might be. We could ask any of the following questions of any given roll of the die:

- What is the numerical outcome?
- What is the square of the numerical outcome?
- How many other possible outcomes are less than the value rolled?
- What is the sum of the top and opposite faces?

Each of these questions corresponds to a distinct **random variable** on the probability space of the die roll. Loosely speaking, a random variable (or RV) is simply a function of the outcome of a random process.

Question 1. Suppose we roll a 5. What are the values of the random variables described in the above items 1-4?

Now suppose we have two fair 6-sided dice, and let *X* be the RV that denotes the sum of the rolled values. What is the probability that *X* > 8? Well, we have the possible outcomes {(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}, so there are 10 outcomes where *X* > 8. But there are 6(6) = 36 total outcomes, so the desired probability Pr[*X* > 8] = 10/36 = 5/18. The idea behind this example is that we can construct a RV and compute an associated probability, because the RV has an associated *probability distribution* of possible values. For instance, we can compute

Pr[*X* = 2] = 1/36; Pr[*X* = 3] = 2/36; Pr[*X* = 4] = 3/36; ….

and in doing so, we have completely specified the probability distribution of *X*.

Question 2. Complete the above list by computing Pr[*X* = *n*] for any real number *n*.

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