Recall that two random variables X and Y are independent if holds for all a,b E
ID: 3070437 • Letter: R
Question
Recall that two random variables X and Y are independent if holds for all a,b E R. This should be covered in your prerequisites, but you can also refer to the Wikipedia page at https://en.wikipedia.org/wiki/Independence_(probability_theory) Assume that X and Y vertible monotonic functions f and g, f(X) and g(Y) are independent. [Note: The condition can be weakened to measurable functions, but in the later case the proof may require certain knowledge from measure theory. are independent random variables. Prove that for any inExplanation / Answer
By definition of independence random variables, Random variables X and Y are said to be independent if (X) is independent of (Y). In other words, X and Y are independent if, for any two borel sets B1 and B2, P[XB1,YB2] = P[XB1]P[YB2].
The -algebra generated by f(X) is a sub--algebra of the -algebra generated by X, and the -algebra generated by g(X) is a sub--algebra of the -algebra generated by Y. And for any borel set B we have composite function (fX)1(B) = X1(f1(B)) and (gY)1(B) = Y1(g1(B))
Since X,Y are independent, therefore, we can get that, X1(f1(B)) and Y1(g1(B)) are independent. Also, since (X) and (Y) are independent therefore (f(X)) and (g(Y)) are also independent. Therefor giving us that f(X) and g(Y) are independent.
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