Proving the Independence Problem in Probability Theory: A Guide

In summary, the conversation is about a problem involving a countable probability triple (Q,F,P) and the existence of an independent sequence (A1,A2,A3,...) with probabilities of 1/2 for each element. The hint suggests proving that for every w in Q and n in N, P({w}) is less than or equal to 1/(2^n), and then using this to derive a contradiction. It is also mentioned that if Q is countable, there is an element \omega with P(\{\omega\}) greater than 0, which can be used in the proof.
  • #1
dottidot
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Independence Problem: Please Help!

I have been trying to figure out the proof to this problem for the past couple of days and still don't have an answer. The question is as follows:

Let (Q,F,P) be a probability triple such that Q is countable. Prove that it is impossible for there to exist a sequence A1,A2,A3,... E F which is independent, such that P(Ai) = 1/2 for eash i. [Hint: First prove that for each w E Q, and each n E N, we have P({w}) <= 1/(2^n). Then derive a contradiction.

Note that Q, represents Omega, F the sigma-algebra/sigma-fiels and E is "an element of or member of"

Would appreciate any help with this.
 
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  • #2


Add this to the hint. If [itex]Q [/itex] is countable, then there is [itex] \omega \in Q[/itex] with [itex]P(\{\omega\}) > 0 [/itex]. So, for large [itex] n [/itex] we have [itex] P(\{\omega\}) > 1/2^n[/itex].
 

1. What is the Independence Problem?

The Independence Problem is a concept in mathematics and computer science that refers to the difficulty in determining whether the results of one experiment are truly independent from the results of another experiment.

2. Why is the Independence Problem important?

Understanding and solving the Independence Problem is crucial in many fields, such as statistics, genetics, and artificial intelligence, where the validity and reliability of experimental results are essential for making accurate predictions and decisions.

3. How is the Independence Problem addressed in research?

Researchers use various statistical methods and experimental designs to reduce the likelihood of dependencies between different experiments. These can include randomization, control groups, and statistical tests for independence.

4. What are some potential consequences of not considering the Independence Problem?

If the Independence Problem is not properly addressed, it can lead to inaccurate conclusions and predictions, which can have serious consequences in fields such as healthcare, finance, and engineering. It can also waste time and resources on experiments that do not provide truly independent results.

5. Is the Independence Problem fully solvable?

The Independence Problem is a complex and ongoing issue in research and there is no single solution that can guarantee complete independence between experiments. However, through careful experimental design and analysis, researchers can minimize the impact of dependencies and improve the reliability of their results.

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