Example 1.4

Consider you toss a coin  number of times. Denote the realization of head as , while that of a tail is denoted as . Then the stochastic process is given as  where . Diagrammatically it is shown below

Figure 1.3: Stochastic process depicting the occurrence of head (H) or tail (T) when a coin is flipped  number of times

Now if  is the function which maps the relation between the sample space and the real line, [0, 1], then we denote it by , i.e., , i.e., . Pictorially it is depicted as shown in Figure 1.4 and Figure 1.5.

Figure 1.4: Probability mapping for a stochastic process from W to

Figure 1.5: Probability mapping for a stochastic process from  Ω to X( , ) and then to R