How I understood the beginning example about the mean: We defined our sample space to be Omega = { [a,0] , [0,b] }. We defined the distribution over this sample space to be uniform (hence the 1/2). Now each sample can be seen as a random variable which takes on one of the values in Omega. By definition, the expectation is the sum of the products of each outcome with its probability. However, if we define a random variable to be the sum of the samples then we get the expectation of the new random variable is sum of the expectations of each sample. Each sample has the same expectation thus we can reduce the sum to a multiplication by s. PS I don't have very advanced knowledge of probability theory (I'm taking a course right now). If I have made a mistake let me know.####[]