3.1 Discrete Random Variables

Let us suppose each independent observation of a stochastic process is a Bernoulli trial and hence can be classified as success or failure; true or false; heads or tails; 1 or 0. Consider the packet storage example in Chapter 1 which is equivalent to drawing at random with replacement from a box containing three pennies, four nickels and a dime. Suppose drawing a dime (or getting a 10 kilobit packet) is considered a success while anything else is a failure. To model this sequence of n independent, identically distributed Bernoulli trials we define the product space ? = { p ₁, p ₂, p ₃, n ₁, n ₂, n ₃, n ₄, d} ⁿ. Let { X _i} ⁿ _{i = 1} be a sequence of random variables such that X _i( ?) = 1 if the i ^th packet associated with the sample point ? ? ? is d; X _i( ?) = 0 otherwise. and P is the equiprobability measure so the marginal distributions are P( X _i = 1) = 1 ? P( X _i = 0) = p = 1/8.

Another example is the model for flipping a coin in Example 2.3. There we saw that to model a sequence of n independent, identically distributed Bernoulli trials we define the product space