Probstat/notes/random variables

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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

Expectations