Probstat/notes/random variables

This is part of probstat.

In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a random variable, which is a function from the sample space to real numbers, to represent the random quantity that we are interested in.

For example, consider the following experiment. We toss two dice. Let a random variable X be the sum of the values of these two dice. The table below shows the outcomes and probabilities related to X.

i	Outcomes for which X = i	Probability P{ X = i }
2	(1,1)	1/36
3	(1,2), (2,1)	2/36
4	(1,3), (2,2), (3,1)	3/36
5	(1,4), (2,3), (3,2), (4,1)	4/36
6	(1,5), (2,4), (3,3), (4,2), (5,1)	5/36
7	(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)	6/36
8	(2,6), (3,5), (4,4), (5,3), (6,2)	5/36
9	(3,6), (4,5), (5,4), (6,4)	4/36
10	(4,6), (5,5), (6,4)	3/36
11	(5,6), (6,5)	2/36
12	(6,6)	1/36

A random variable X also induces events related to it. From the previous example, the event that X=10 corresponds to the subset {(4,6), (5,5), 6,4)} of the sample space. Also, if the event X >= 11 corresponds to {(5,6), (6,5), (6,6)}.

Therefore, it is reasonable to consider the probability of events defined by random variables. From the two-dice example, we have P{ X >= 11 } = P({(5,6), (6,5), (6,6)}) = 3/36.

Given a random variable X, a probability mass function p of X is defined as p(i) = P{ X = i }. We usually denote the probability mass function as pmf.

Another example

Suppose that we pick two numbers from the set {1,2,3,4} without replacement. Let Y be the larger number. The following table shows each events defined on various values of Y.

i	Outcomes	Probability P{ Y = i }
1	-	0
2	(1,2), (2,1)	2/12
3	(1,3), (2,3), (3,1), (3,2)	4/12
4	(1,4), (2,4), (3,4), (4,1), (4,2), (4,3)	6/12