Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta

probabilities , respectively. Since are disjoint and their union is equal to the entire sample space S, we have

      (3.7.1)

      We can arrange the values of and the corresponding probabilities in table form as follows:

      (3.7.2)

      The values of the discrete random variable together with their associated probabilities are called the discrete distribution of . The function , defined by

      (3.7.3)

      is called the probability function (p.f.) of . Ordinarily, we drop the e and refer to the random variable as X. The set of possible values is called the sample space of the random variable X.

      

      Example 3.7.1 (Defining concept of the probability function) Let X be a random variable denoting the sum of the number of dots that appear when two dice are thrown. If each of the 36 elements in the sample space is assigned the same probability, namely , then , the probability function of X, is as follows:

	2	3	4	5	6	7	8	9	10	11	12
	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

      If the sample space S has an infinite number of elements and if the random variable x can take on a countably infinite set of values we have a discrete random variable with sample space {}.

      Example 3.7.2 (Probability function for an event to occur) Let X be a random variable denoting the number of times a die is thrown until an ace appears. The sample space of X is and the probability function is given by the table:

	1	2
	1/6

      since

      Note that the probability function must possess the following properties.

If any one of these properties does not hold, then is not a probability function.

      We conclude this section with the comment that in this section, we have discussed only discrete random variables. There is indeed another type of random variables, called continuous random variables, discussed extensively in Chapter 5. Suffice it to say here that a continuous random variable may take all values in at least one interval, and it, of course, contains an infinite number of values that are not countable. This is in contrast with a discrete random variable, which takes values that are countable, as discussed here and in Chapter 4.

Review Practice Problems

      1 Certain pieces made by an automatic lathe are subject to three kinds of defects X, Y, Z. A sample of 1000 pieces was inspected with the following results: 2.1% had type defect, 2.4% had type defect, and 2.8% had type defect. 0.3% had both type and type defects, 0.4% had both type and type defects, and 0.6% had both type and type defects. 0.1% had type , type , and type defects.Then find:What percent had none of these defects?What percent had at least one of these defects?What percent were free of type and/or type defects?What percent had not more than one of these defects?

      2 Two inspectors A and B independently inspected the same lot of items. Four percent of the items are actually defective. The results turn out to be as follows: 5% of the items