alt="equation"/>
The values of the discrete random variable
(3.7.3)
is called the probability function (p.f.) of
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
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2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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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
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
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1 | 2 |
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1/6 |
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since
Note that the probability function
If any one of these properties does not hold, then
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 are called defective by A, and 6% of the items are called defective by B. 2% of the items are correctly called defective by A, and 3% of the items are correctly called defective by B. 4% of the items are called defective by both A and B, and 1% of the items are correctly called defective by both A and B.Make a Venn diagram showing percentages of items in the eight possible disjoint classes generated by the classification of the two inspectors and the true classification of the items.What percent of the truly defective items are missed by inspectors?
3 A
