Question 1)

Note: iid=idependent and identically dsitributed

I don't get the part underlined in red. Why are they independent? How can we prove that, or at leastintuitively make senseof it?

Question 2) If X1,X2,...,X6 are independent random varaibles, then my textbook has a theorem saying that g1(X1), g2(X2), ..., g6(X6) are also independent, where the gi's are any function of a single random variable.

But how about f1(X1,X2,...,X5), f2(X6)? If X1,X2,...,X6 are independent, are any function of X1,...,X5 and any function of X6 independent?

Any help is greatly appreciated!