Similarly, when we assert that two random variables are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other. For instance, the height of a person and their IQ are independent random variables. Another typical example of two independent variables is given by repeating an experiment: roll a die twice, let X be the number you get the first time, and Y the number you get the second time. These two variables are independent.
We define two events E_{1} and E_{2} of a probability space to be independent iff
If P(E_{2}) ≠ 0, then the independence of E_{1} and E_{2} can also be expressed with conditional probabilities:
If we have more than two events, then pairwise independence is insufficient to capture the intuitive sense of independence. So a set S of events is said to be independent if every finite nonempty subset { E_{1}, ..., E_{n} } of S satisfies
This is called the multiplication rule for independent events.
We define random variables X and Y to be independent if
If X and Y are independent, then the expectation operator has the nice property
Furthermore, if X and Y are independent and have probability densities f_{X}(x)and f_{Y}(y), then (X,Y) has a joint density of
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