cribed first in an equation, then in words, then in a graph, and, finally (phew), using some intuition. (1) In an equation (3.2) A Review of Some Important Math, Stats, and Systems (2) In words, if you want to know how likely it is that tomorrow's temperature x will be between 70 degrees and 80 degrees, all you have to do is integrate p(x) over the range of 70 to 80. (3) In a graph, an example of a possible p(x) is shown in Figure 3.1. If you want to figure out the probability that tomorrow's temperature X will be between 70 and 80, all you have to do is figure out the area under the p(x) curve between 70 and 80. (4) And now, we turn on the intuition. Intuitively speaking, p(x) at x = 70 gives you an idea how likely it is that tomorrow's temperature will be about 70 degrees. Values of x at which p(x) is biggest are those values most likely to happen. 70 75 80 Figure 3.1 Possible p(x) for X= tomorrow's temperature 3.1.4 The Mean and the Variance If you tell me F(X) or p(x), then I know everything there is to know about x, but what if I don't want to know everything about X, or what if it's hard for you to tell me everything about X. In either case, there are some common ways for providing partial (but important) information about X. Chapter Three (1) the mean, Xm (also known as E(x)): One thing you can tell me is the average (or mean) value of X. If X is tomorrow's temperature, then X m is the average (considering all the years gone by), of tomorrow's temperature. If you know p xx, then you can easily compute Xm like this: (3.3) (2) the variance, 02 : Another important piece of information about the random variable x is how much x varies. That's measured by a well-named term, variance. If x changes a lot (for example, if tomorrow's temperature could be anywhere between 40 and 120 degrees, and you really can't be sure where in that range it will fall), then the variance is a big number. If X changes very little (for example, you're very sure tomorrow's temperature will be between 73 and 77 degrees), then the varian