Unformatted text preview: X be the number showing for a single roll of a fair die. Then pX (i) =
integers i with 1 ≤ i ≤ 6. 21 1
6 for 22 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES Example 2.1.3 Let S be the sum of the numbers showing on a pair of fair dice when they are
rolled. Find the pmf of S.
Solution: The underlying sample space is Ω = {(i, j ) : 1 ≤ i ≤ 6, 1 ≤ j ≤ 6}, and it has 36
1
possible outcomes, each having probability 36 . The smallest possible value of S is 2, and {S =
1
2} = {(1, 1)}. That is, there is only one outcome resulting in S = 2, so pS (2) = 36 . Similarly,
2
3
{S = 3} = {(1, 2), (2, 1)}, so pS (3) = 36 . And {S = 4} = {(1, 3), (2, 2), (3, 1)}, so pS (4) = 36 , and
so forth. The pmf of S is shown in Fig. 2.1.
pS
1/6 2 3 4 5 6 7 8 9 10 11 12 Figure 2.1: The pmf of the sum of numbers showing for rolls of two fair dice. Example 2.1.4 Suppose two fair dice are rolled and that Y represents the maximum of the two
numbers showing. The same set of outcomes, Ω = {(i, j ) : 1 ≤ i ≤ 6, 1 ≤ j ≤ 6}, can be used as in
Example 2.1.3. For example, if a 3 shows on the ﬁrs...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
 Spring '08
 Zahrn
 The Land

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