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Summary: Conditional distributions. The continuous case.
Suppose Y is a continous random vector.
For any event E we let
FE,Y (y) = P(E {Y y}), y R.
We let
fE,Y = FE,Y
with the understanding that the domain of fE,Y is the set of y R such that FE,Y exists at y. It follows
from real variable theory that fE,Y is locally integrable. Since
m
i=1
FE,Y (yi) - FE,Y (yi-1) == P(E {y0 < Y ym}) P(y0 < Y ym) =
ym
y0
fY (y) dy
whenever - < y0 y1 · · · ym < it follows from real variable theory that
(1) FE,Y (y) =
y
-
fE,Y (w) dw, y R.
Note that if y R then P(E|Y = y) is undefined because P(Y = y) = 0 since Y is continous. Let us
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