19
Random Variables and Distributions
Having a probability space to model our experiments and observations is fine and all, but in almost all of the cases, we are interested in a quantitative measure of the outcome. To give you an example, let’s consider an already familiar situation: tossing coins. Suppose that we are tossing a fair coin n times but we are only interested in the number of heads. How do we model the probability space this time?
By taking things one step at a time; first, we construct an event space by enumerating all possible outcomes in a single set, just like we already did in Section 18.2.1:
Since the coin is fair, each outcome ω has the probability P(ω) = 
. This probability space (Ω,Σ,P) is nice and simple so far. Using the additivity of probability measures (see Definition 77), we can calculate the probability of any event. That is, for any A ∈ Σ, we have
where j â‹…j denotes the number of elements in...
