Combined sample spaces .

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In summary, the conversation discusses the relationship between measure theory and probability theory, particularly in regards to combining multiple sample spaces and extracting the distribution of the new sample space from previously known distributions. The concept of independence from irrelevant alternatives in decision theory is also mentioned. The general case is represented by the equation P\{x\} = \sum_{k=1}^N P\{x|k\}P\{k\}, where N is the number of spaces.
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Dear all

I've just begun studying measure theory , and i can't help it but to think of it in terms of probability theory , i don't know if that is right or wring . any way , i have this naive question :

consider the following : we have n sample spaces [tex]\Omega_{}i[/tex], each with a distribution P[tex]_{}i[/tex] ( i=1,...n) , if we combine (union) the sample spaces to form a new sample space whose distribution is unknown , is there a way to extract the distribution of the new sample space from the previously know distributions ??
 
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  • #2
The simplest case is to assume disjoint sample spaces. In that case, the probability of obtaining element x from the k'th space, x(k), will be P{x(k)} = P{x|k}P{k} = Pk{x}P{k}, where P{k} is the probability of obtaining the k'th space within the set of all spaces, or the measure of the k'th space in the union.

In general:

[tex]P\{x\} = \sum_{k=1}^N P\{x|k\}P\{k\}[/tex]

where N is the number of spaces.

This is related to the axiom of independence [from] irrelevant alternatives in decision theory.
 
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  • #3
what about the general case ?
 
  • #4
See the portion of my post that begins with "In general."
 

What is a combined sample space?

A combined sample space is a mathematical concept used in probability theory to represent all possible outcomes of two or more random events occurring together. It is the union of all the individual sample spaces for each event.

How do you calculate the size of a combined sample space?

The size of a combined sample space is calculated by multiplying the number of outcomes for each individual event. For example, if event A has 3 possible outcomes and event B has 4 possible outcomes, the combined sample space will have 3 x 4 = 12 possible outcomes.

What is the difference between a combined sample space and a joint sample space?

While a combined sample space represents the union of all possible outcomes for two or more events, a joint sample space represents the intersection of these outcomes. In other words, a joint sample space only includes outcomes that satisfy all of the events.

What is the significance of combined sample spaces in probability theory?

Combined sample spaces are important in probability theory as they allow us to analyze the likelihood of multiple events occurring together. By considering all possible outcomes in a combined sample space, we can determine the probability of specific combinations of events happening.

Can a combined sample space contain an infinite number of outcomes?

Yes, a combined sample space can contain an infinite number of outcomes. This is often the case in continuous probability distributions, where the sample space is represented by a range of values rather than a discrete set of outcomes.

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