The following is the definition of a random variable:Let ( Ω , F , P ) be a probability space. A random variable is a real-valued function X on Ω such that for all x ∈ R , { ω : X ( ω ) ∈ B } ∈ F , ∀ B ∈ B ( R )I don't understand how this makes sense if our choice of F can be arbitrary.Let's say we are rolling a biased dice such that each side has a different probability, and the information we know about the system is F = { ∅ , { 1 } , { 2 , . . . , 6 } , Ω } -- then for this F, according to the definition, X is not a random variable (because you could construct a (disjoint) Borel set which covers the probabilities of e.g. { 1 , 3 , 5 }, and { 1 , 3 , 5 } is not in F). How does this make sense?

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The following is the definition of a random variable:Let   (  Ω  ,      F    ,  P  ) be a probability space. A random variable is a real-valued function X on   Ω such that for all   x  ∈      R    ,  {  ω  :  X  (  ω  )  ∈  B  }  ∈      F    ,      ∀    B  ∈  B  (      R    )I don&#039;t understand how this makes sense if our choice of   F can be arbitrary.Let&#039;s say we are rolling a biased dice such that each side has a different probability, and the information we know about the system is   F  =  {  ∅  ,  {  1  }  ,  {  2  ,  .  .  .  ,  6  }  ,  Ω  } -- then for this   F, according to the definition,   X is not a random variable (because you could construct a (disjoint) Borel set which covers the probabilities of e.g.   {  1  ,  3  ,  5  }, and   {  1  ,  3  ,  5  } is not in   F). How does this make sense?

jarakapak7 · Accepted Answer

Well sure, then the biased dice roll would not be a random variable in the sense you&#039;re used to thinking about it. However, it is quite meaningless to claim that it should be a traditional random variable in the first place, since       F  , or often:       F   or   Σ, is (intuitively) the sigma-algebra of possible measurable events which we can consider.With your choice of       F  , only the events: nothing, 1, {2⋯6} or everything, can be considered in terms of probability. It is meaningless to say that the probability of 3 is different to that of 5, since neither 3 nor 5 are measurable events and as such do not have any sense of probability. You can only ask questions such as: what is the probability   X  ≥  2, or the probability   X  =  1. You cannot talk of the probability   X  ≥  3 even, since   {  3  ,  4  ,  5  ,  6  } is not a measurable event and as such has no probability defined for it.So: yes, the sigma-algebra of events is arbitrary, but so is the probability space itself. You keep the theory general, and make specific definitions and examples when it suits.

The following is the definition of a random variable: Let ( <mi mathvariant="normal">Ω<

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