206 CHAPTER 5 Discrete Probability Distributions DEFINITIONS A discrete random variable has a collection of values that is finite or countable. (If there are infinitely many values, the number of values is countable if it is possible to count them individually, such as the number of tosses of a coin before getting heads.) A continuous random variable has infinitely many values, and the collection of values is not countable. (That is, it is impossible to count the individual items because at least some of them are on a continuous scale, such as body temperatures.) This chapter deals exclusively with discrete random variables, but the following chapters deal with continuous random variables. Probability Distribution: Requirements Every probability distribution must satisfy each of the following three requirements. 1. There is a numerical (not categorical) random variable x, and its numerical values are associated with corresponding probabilities. 2. ΣP1x2 = 1 where x assumes all possible values. (The sum of all probabilities must be 1, but sums such as 0.999 or 1.001 are acceptable because they result from rounding errors.) 3. 0 … P1x2 … 1 for every individual value of the random variable x. (That is, each probability value must be between 0 and 1 inclusive.) The second requirement comes from the simple fact that the random variable x represents all possible events in the entire sample space, so we are certain (with probability 1) that one of the events will occur. The third requirement comes from the basic principle that any probability value must be 0 or 1 or a value between 0 and 1. Births TABLE 5-2 Probability Distribution for the Number of Females in Two Births x: Number of Females in Two Births P (x) 0 0.25 1 0.50 2 0.25 For the purposes of this example, assume that male births and female births are equally likely. [In reality, P1male birth2 = 0.512.] Let’s consider two births, with the following random variable: x = number of females when two babies are born The above x is a random variable because its numerical values depend on chance. With two births, the number of females can be 0, 1, or 2, and Table 5-2 is a probability distribution because it gives the probability for each value of the random variable x and it satisfies the three requirements listed earlier: 1. The variable x is a numerical random variable, and its values are associated with probabilities, as in Table 5-2. 2. ΣP1x2 = 0.25 + 0.50 + 0.25 = 1 3. Each value of P1x2 is between 0 and 1. (Specifically, 0.25 and 0.50 and 0.25 are each between 0 and 1 inclusive.) The random variable x in Table 5-2 is a discrete random variable, because it has three possible values 10, 1, 22, and three is a finite number, so this satisfies the requirement of being finite or countable. EXAMPLE 1 YOUR TURN. Do Exercise 7 “Plane Crashes.”
RkJQdWJsaXNoZXIy NjM5ODQ=