Let us say that we have some set of independent random variables: X = { X i } i = 1 n defined over a probability space of ( Ω , F , P ). I want to understand whether the following holds:If F X = σ ( { X 1 , X 2 , … , X n } ) , then is it, in general, true that: | F X | = ∑ ∀ i ∈ I n | σ ( { X i } ) | ?As we know, as X consists of independent R.V.-s, then we can state that ∀ i , j ∈ I n : σ ( { X i } ) and σ ( { X j } ) are independent. But how to proceed from this fact to the split of the cardinality of F X ?I would appreciate any help, thank you in advance!

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Let us say that we have some set of independent random variables:   X  =  {      X    i        }          i      =      1        n   defined over a probability space of       (  Ω  ,      F    ,  P  ). I want to understand whether the following holds:If             F        X    =  σ      (          {              X        1            ,              X        2            ,      …      ,              X        n            }        )  , then is it, in general, true that:       |                  F            X        |    =      ∑          ∀      i      ∈              I        n                  |    σ  (  {      X    i    }  )      |  ?As we know, as   X consists of independent R.V.-s, then we can state that   ∀  i  ,  j  ∈      I    n    :  σ  (  {      X    i    }  ) and   σ  (  {      X    j    }  ) are independent. But how to proceed from this fact to the split of the cardinality of             F        X  ?I would appreciate any help, thank you in advance!

Jaylee Dodson · Accepted Answer

It should not be that hard to show then that in general the proposed cardinality relation does not hold!

Let us say that we have some set of independent random variables: X = <mo fence="false" st

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