This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1.2. DISCRETE PROBABILITY DISTRIBUTIONS 29 otherwise. Infinite sample spaces require new concepts in general (see Chapter 2), but countably infinite spaces do not. If Ω = { ω 1 , ω 2 , ω 3 , . . . } is a countably infinite sample space, then a distribution function is defined exactly as in Definition 1.2, except that the sum must now be a convergent infinite sum. Theorem 1.1 is still true, as are its extensions Theorems 1.2 and 1.4. One thing we cannot do on a countably infinite sample space that we could do on a finite sample space is to define a uniform distribution function as in Definition 1.3. You are asked in Exercise 20 to explain why this is not possible. Example 1.13 A coin is tossed until the first time that a head turns up. Let the outcome of the experiment, ω , be the first time that a head turns up. Then the possible outcomes of our experiment are Ω = { 1 , 2 , 3 , . . . } . Note that even though the coin could come up tails every time we have not allowed for this possibility. We will explain why in a moment. The probability that heads comes up on the first toss is 1/2. The probability that tails comes up on the first toss and heads on the second is 1/4. The probability that we have two tails followed by a head is 1/8, and so forth. This suggests assigning the distribution function m ( n ) = 1 / 2 n for n = 1, 2, 3, .. . . To see that this is a distribution function we must show that...
View Full
Document
 Spring '09
 scf

Click to edit the document details