Probability distributions

The probability distribution function (PDF) that is induced by a random variable X is the function f_X, defined by:

Here, the expression X = x means the event of all outcomes e for which X(e) = x.

Returning to our coin example, let's compute the probability distribution f_X. It is defined on the range of X, which is the set {0, 1, 2, 3, 4}. For example:

In fact, the probability distribution f_X for the first version of the coin example is precisely the same as the p(s) function in the second version, tabulated in Table 4-1.

The properties of a probability distribution follow directly from those governing probabilities. They are:

• 0 ≤ f(x) ≤ 1, for every x ∈ X(S)

• ∑ f(x) = 1

Here is another classic example. The experiment is to toss two balanced dice, one red and one green, and observe the two numbers represented by the dots showing on top. The sample space S has 36 elements:

If the dice are balanced, then each one of these 36 possible outcomes has the same probability...