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Stats - find the distribution function of an infinite sample space.

by dtsar
Tags: sample space, stats
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dtsar
#1
Dec13-11, 03:01 PM
P: 1
1. The problem statement, all variables and given/known data

A die is rolled until the first time that a six turns up. We shall see that the
probability that this occurs on the nth roll is (5/6)n−1 (1/6). Using this fact,
describe the appropriate infinite sample space and distribution function for
the experiment of rolling a die until a six turns up for the first time. Verify
that for your distribution function the Ʃ(ω)=1, as ω→∞

2. Relevant equations

Relevant equations are in the question.


3. The attempt at a solution
Ω = { 6, N6, NN6, NNN6, ... , N...N6 }
Also, I know that the equation is exponential decay, but I just don't know how to get the formula...
I also know that it adds up, because (1/6)+(5/6)(1/6) + (5/6)^2(1/6) .. etc eventually equal to 1 as N→∞
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Ray Vickson
#2
Dec13-11, 05:23 PM
Sci Advisor
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P: 5,086
Quote Quote by dtsar View Post
1. The problem statement, all variables and given/known data

A die is rolled until the first time that a six turns up. We shall see that the
probability that this occurs on the nth roll is (5/6)n−1 (1/6). Using this fact,
describe the appropriate infinite sample space and distribution function for
the experiment of rolling a die until a six turns up for the first time. Verify
that for your distribution function the Ʃ(ω)=1, as ω→∞

2. Relevant equations

Relevant equations are in the question.


3. The attempt at a solution
Ω = { 6, N6, NN6, NNN6, ... , N...N6 }
Also, I know that the equation is exponential decay, but I just don't know how to get the formula...
I also know that it adds up, because (1/6)+(5/6)(1/6) + (5/6)^2(1/6) .. etc eventually equal to 1 as N→∞
(i) What you wrote was (5/6)n-1 - 1.(1/6), which means (5n/6) - (1/6). You should have written either (5/6)^(n-1) - (1/6) or used the "S U P" button to get (5/6)n-1 - (1/6).
(ii) Saying Ʃ(ω)=1, as ω→∞ makes no sense: the ω need not be numbers, so they can't "go to infinity". Just saying Ʃ(ω)=1 is enough.
(iii) You write Ω as though it has an "end", but it doesn't just write Ω = {6, N6, NN6, NNN6, ... }. Also, if you use N here you should not later say "as N → ∞". Use a different symbol.

Aside from these writing issues, I don't see your problem; you seem to have answered the questions you were asked. For example, when you say "I just don't know how to get the formula...", that is not relevant: you are *given* the formula, and are asked to use it. You have done that correctly.

RGV


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