prob.part18.37_38 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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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...
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prob.part18.37_38 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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