heit) than it is to be 0 degrees. This section is all about how you can describe a random variable, like tomorrow's temperature, to someone without using a lot of words. Chapter Three One way to fully characterize our random variable x is by a function called the probability distribution function, Fx(X). The function F(X) is defined in words as follows: Fx(X) is the likelihood that the random variable x is less than the number X. In a nice, neat equation, Fx (X) is defined as F,(X)=P(x<X) (3.1) where P(_) is shorthand for the words "probability that _happens." Let me clarify the meaning of Fx (X) by an example. Let's again go back to x being tomorrow's temperature, and let's say it's the middle of summer. Then: (1) Fx (0) = P(x<0) = (the probability that the temperature is less than 0), and this may be 1/1000000 (1 in a million), and (2) Fx (70) = P(x<70) = (the probability that the temperature is less than 70), and this may be 1/2 (1 in 2). By providing FX()) for all possible values of X, you completely characterize your random variable. Here are four simple properties of FX() (1) 0<F((X) <1: that is, since FX()) represents the probability that X <X, it, like all probabilities, must be between 0 (never happens) and 1 (always happens). (2) F(-oo)=0: that is, Fx (-00) = P(x < - oo) = (the probability that x is less than -00) =0 (since no number can be smaller than -00) . (3) Fx( 00) =1: that is, Fx (00) = P(x < 00) = (the probability that x is less than oo) = 1 (since every value must be smaller than oo). (4) Fx(x)) F(x) ifx,>x2: that is, for example, the probability that X is less than 20 is at least as big as the probability that x is less than 10. 3.1.3 The Density Function: A Second Way to Describe X A second way to describe our random variable x is to use a different function called the probability density function (pdf for short). Let's take a look again at our random vari- able x, which represents tomorrow's summer temperature. The pdf for this variable is denoted Px (x) or p(x), and its meaning will be des