406 17 Describing Random Signals I I I e I I Figure 17.1: Example of sample functiurls ,( t ) Under certain pre-conditions, that we will deal with later in niorc dctail, the enscnible average can also be expressed by the time-average for a sample function Example 17.1 We use the example of a die to show the three methods for finding an exact ox approximate expected value: Fkom howledge of the process, which is the complete symirictry of the die, we can say that every number has the same prob;thility of appearing, so thc expectcd v:tlur is: E { z ( t ) }= 14-2 + + + + = 3.5 * As the die's syrrirrietry does riot change, the expected valiie any time t e is the same for If wc repeat the procedure with ~ n a n ydice whose syrnrnetxy properties we riot know precisely, wc can also approximately calculate the eiisenible average by averaging all of the iiurrhm on tlie die's faces As the expected values can in general change over time, we should take the ensemble Iiiean for all dice at, the sairie time