- #1
the4thamigo_uk
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Ive been working with random variables for a while and only today have I come up with a basic question that undermines what I thought I knew...
If I have two random variables X and Y, when am I allowed to multiply them? i.e. Z=XY
Let S_1 and S_1 be sigma algebras such that S_1 is contained in S_2
Cases
i) X and Y are both S_1 measurable
It seems clear that Z=XY exists and is also S_1 measurable
ii) X is S_1 measurable and Y is S_2 measurable
In this case X is also S_2 measurable, but Y is not S_1 measurable. (Am I correct to say this?)
Can we form Z=XY and if so does Z simply become S_2 measurable?
iii) Assume S_3 is not a subset of either S_1 or S_2
Can we write Z=XY?Thanks for your help
If I have two random variables X and Y, when am I allowed to multiply them? i.e. Z=XY
Let S_1 and S_1 be sigma algebras such that S_1 is contained in S_2
Cases
i) X and Y are both S_1 measurable
It seems clear that Z=XY exists and is also S_1 measurable
ii) X is S_1 measurable and Y is S_2 measurable
In this case X is also S_2 measurable, but Y is not S_1 measurable. (Am I correct to say this?)
Can we form Z=XY and if so does Z simply become S_2 measurable?
iii) Assume S_3 is not a subset of either S_1 or S_2
Can we write Z=XY?Thanks for your help