If for are , then can we show that for are also ?
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yup Whatever you had for the distribution of X, your new random variables will all have that new distribution. And also if is independent of for then is independent of for
is there a way to show that are independent?
do you want a measure theoretical proof or for just continuous or discrete rvs?
any one of those is fine. thank you very much
Think about the independence of the sigma-algebras generated by each of the random variable
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