mike s.

Dear Group,

Let X and Y be continuous random variables and let g(x) be smooth. Let
f_XY(x,y), the joint density of X and Y, exist everywhere.
Is it true that

Prob( g(X) + Y <= y GIVEN X = x ) = Prob( g(x) + Y <= y ) ?

Could you sketch a proof or give a counter example or a pointer to the
literature? What if X and Y are mutually independent vectors of random
variables?

TIA,

Mike S.

g. a. edgar

Not without some independence. (If I understand the question.)
Say X and Y have joint density m(x,y) = x+y on
[0,1] x [0,1]. That has double integral 1, right?
Let g(x) = x. Then

P( X + Y <= 1 | X = 2/3)
= (integral(y=0 to 1/3) m(2/3,y) dy)/(integral(y=0 to 1) m(2/3,y) dy)
= 5/21 , but

P( 2/3 + Y <= 1)
= doubleintegral((x=0 to 1)(y=0 to 1/3) m(x,y) dx dy) = 2/9 .


If X and Y are independent, I think it is correct.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/