Probability and Statistics II Week 1

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical process and the resistance of a thermocouple. More generally a random variable is a rule for assigning a particular number to a particular experimental outcome. For example, if the result of an experiment is recorded as say ‘the experiment was successful’ or ‘the experiment was not successful’, then to obtain a random variable we must code the results so that, for example, a success corresponds to 1 and failure to 0. A random variable is denoted by , where corresponds to events Ei respectively. Definition 1.1 If S is a sample space with a probability measure and X is a real – valued function defined over the elements of S, then X is called a random variable. We shall always denote random variables by capital letters and their values by the corresponding lowercase letters; for instance, we shall write x to denote a value of the random variable X. Example 1.1 Consider an experiment in which we roll a pair of dice and considering the sum of upturned faces. Let X be the sum of upturned faces when the two dice are tossed. Let us assume that each of the 36 possible outcomes has the probability . The sample space of this experiment is; 1 2 3 4 5 6 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) We attach a number to each point: for instance we attach a number 2 to the point (1, 1), the number 6 to the point (1, 5), the number 8 to the point (6, 2), and so forth. Evidently, we associated with each point the value of a random variable, that is, the corresponding total rolled with the pair of dice. Similarly, we observe that the random variable X takes on the value 9, and we write X = 9 for the subset of the sample space S. Thus, X = 9 is to be interpreted as the set of elements of S for which the total is 9 and, more generally, X = x is to be interpreted as the set of elements of the sample space for which the random variable X takes on the value x. Example 1.2 Two socks are selected at random and removed in succession from a drawer containing five brown socks and three green socks. List the elements of the sample space, the corresponding probabilities, and the corresponding values w of the random variable W, where W is the number of brown socks selected. Solution If B and G stand for brown and green, the probabilities for BB, BG, GB, and GG are, respectively; , , , and , and the results are shown in the following table: Element of Sample space Probability w BB 2 BG 1 GB 1 GG 0 Also, we can write , for example, for the probability of the event that the random variable W will take on a value 2.