which is a function of the random variable x, which takes the discrete values Xi (i = 1, , N). When writing p(xi), this conceptually means "the probability that event x takes the value Xi." The mean, which is noted <x>, or also x, or E(x), is given by the weighted sum (2.1) 2.1 Mean and variance The mean represents what is also called the expectation value, hence the notation E(x). Contrary to what the word "expectation" would suggest, E(x) is not the value one should expect for the outcome of event x after some sufficient number of trials. Rather, it is the mean value of X that one should expect to become, independent of the number of trials N, as this number indefinitely increases, and assuming that the PDF is constant over time. As an illustration, take the event space S = {1, 2, 3, 4, 5, 6} corresponding to the outcomes of rolling a single die. As we know, the PDF is p(x) = 1/6 for all events X. The mean value is, therefore, (2.2) This example illustrates what should be understood by "expected value." In no way would a die fall on X = 3.5, and nobody would "expect" such a result! The number simply represents the average count of spots, were we to perform an infinite (say, sufficiently large) number of die rolls. Looking back at Fig. 1.2 (counting spots from two dice), we observe that the PDF is centered about x = 7, which represents the mean value of all possibilities. The mean value should, therefore, not be interpreted as representing the event of highest probability, i.e., where the PDF exhibits a peak value, although this might be true in several cases, like the PDF in Fig. 1.2. But the mean does not tell the whole story about a PDF. I introduce a second parameter called PDF variance. The variance, typically noted o2, is defined as the expected value of (x - <x>)2. Thus, using basic algebra: o2=<(x-<r>)? =<x2-2x<x>+<x>2> = <x2>-2<x><x>+<<x>2> (2.3) <x2>-2<r>2+ = xx2>-<x>2 (noting that <<x>2>= <x>2, since <x>2 is a number, not a variable). The variance is the difference between the expected value of x2 and