-- CHAPTER TWO random variables Often we have reasons to associate one or more numbers (in addition to probabilities) with each possible outcome of an experiment. Such numbers might correspond, for instance, to the cost to us of each experi- mental outcome, the amount of rainfall during a particular month, or the height and weight of the next football player we meet. This chapter extends arid specializes our earlier work to develop effective methods for the study of experiments whose outcomes may be described numerically.2-1 Random Variables and Their Event Spaces For t'he study of experinlents whose outconles may be specified numeri- cally, we find it useful to introduce the following definition: A random varzhble is defined by a function which assigns a value of the ESZ rarldonl variable to each sample point in the sample space of an experiment. Each performance of the experiment is said to generate an experimental calzce of the randonl variable. This experinlental value of the random variable is equal to the value of the randonl variable assigned to the sample point which corresponds t-o the resulting experimental outcome. Consider the following example, which will be referred to in several sections of this chapter. Our experiment consists of three independent flips of a fair coin. We again use the notation Event : (F!:,"] on the nth Hip We may define any number of random variables on the sample space of this experiment. We choose the following definitions for two ran- dom variables, h and r: h = total number of heads resulting'from the three flips r* = 1cngt.h of Iongest run resulting from the three flips (a run is a set of successive flips all of which have the same outcome) We now prepare a fully labeled sequential sample space for this experiment. We include the branch traversal conditional probabilities! t,he probability of each experinlental outcon~e, and the values of random variables h and 1. assigwd to each salnple point. The resulting sample spaw is shown at the top of the following page. If this experiment were performed once and the experimental out.conle were the event H1T2T3, we would say that, for this per- formance of the experiment, the resulting experimental values of ran- dom variables h and I. were 1 and 2, respectively. Although we may require the full sample space to describe the dehiled probabilistic structure of an experiment, it may be that our only practical interest in each performance of the experiment will relate to t,he resulting experimental values of one or more random variables. RANDOM VARIABLES AND THEIR EVENT Sample points " "lH2*3 " Hl *2 T3 " qH3 "1 T2 T3 " H2H3 H2T3 oTlI;H3 ' ? TP3 SPACES 43 When this is the case, we may prefer to work in an event space which distinguishes among outcomes only in terms of the possible experi- mental values of the random variables of interest. Let's consider this for the above example. Suppose that our only interest in a performance of the experi- ment has to do with the resulting experimental value of random varia- ble h. We might find it desirable to work with this variable in an event space of the form The four event points marked along the ho axis form a mutually exclu- sive collectively exhaustive listing of all possible experimental outcomes. The event point at any h corresponds to the event "The experimental value of random variable h generated on a performance of the experi- ment is equal to ho," or, in other words, "On a performance of the experiment, random variable h takes on experimental value ho." Similarly, if our concern with each performance of the experi- ment depended only upon the resulting experimental values of random variables h and r, a simple event space would beRANDOM VARIABLES THE PROBABILITY MASS FUNCTION 45 to all possible experiment.al values of the random variable. One such event space could be This event point represents the event "exactly two heads resulted from the three flips and no pair of consecutive flips had the same outcomen Each point in this event space corresponds to the event "On a per- formance of the experiment, the resulting experimental value of random variable x is equal to the indicated value of xo." We next define a function on this event space which assigns a An event point in this space with coordinates ho and 1.0 corresponds to the event "On a performance of the experiment, random variables h and 1, take on, respectively, experimental values ho and T~." The proba,- probability to each event point. The function p,(xo) is known as the probability mass function (PAIF) for discrete random variable x, defined by = bility assignment for each of these six event points may, of course, be --obtained by collecting these events and their probabilities in the origi- nal sequential sample space. The random variables discussed in our example could take on p,(xo) = probability that the experimental value of random variable x obt.ained on a performance of the experiment is equal to xo ZZS ZZZ-= -only experimental values selected from a set of discrete numbers. Such = random variables are known as discrete random variables. Random variables of another type, known as continuous random variables, may take on experimental values anywhere within continuous ranges. Examples of continuous random variables are the exact instantaneous We often present the probability mass function as a bar graph drawn over an event space for the random variable. One possible PMF is sketched below: voltage of a noise signal and the precise reading after a spin of an infinitely finely calibrated wheel of fortune (as in the last example of Sec. 1-2). Formally, the distinction between discrete and continuous ran- dom variables can be avoided. Rut the development of our topics is easier to visualize if we first become familiar with matters with regard to discrete random variables and later extend our coverage to the con-tinuous case. Our discussions through Sec. 2-8 deal only with the discrete case, and Sec. 2-9 begins the extension to the continuous case. Since there must be some value of random variable x associated with every sample point, we must have 1 2-2 The Probability Mass Function We have learned that a random variable is d-efined by a function which assigns a value of that random variable to each sample point. These and,
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