Expectation of a Random Variable/ Part1

The expectation of a random variable is essentially the average value it is expected to take on. Therefore, it is calculated as the weighted average of the possible outcomes of the random variable, where the weights are just the probabilities of the outcomes. As a trivial example, consider the (discrete) random variable X (outcomes of some probabilistic experiment) whose sample space is the set {1,2,3} with probability function given by p(1)=0.3, p(2)=0.1 and p(3)=0.6. If we repeated this experiment 100 times, we would expect to get about 30 occurrences of X=1, 10 of X=2 and 60 of X=3. The average X would then be ((30)(1)+(10)(2)+(60)(3))/100 = 2.3. In other words, (1)(0.3)+(2)(0.1)+(3)(0.6). This reasoning leads to the defining formula: