I need advice on this problem. It is related to Simpson's Paradox.
Consider three binary variables $X, Y, Z$ and all taking values in $\{0, 1\}.$
Consider the following inequalities:
$P(X = 1) > P(Y = 1)\tag 1$
$P(X = 1 \land Z = 1) < P(Y = 1 \land Z = 1)\tag 2$
$P(X = 1 \land Z = 0) < P(Y = 1 \land Z = 0) \tag 3$
Assumed all inequalities, (1) (2) and (3), are true, which of the following claims are true?:
a) If I perform a linear regression of X on Z, the $e^Z$ must be negative.
b) If I perform a linear regression of Y on Z, the $e^Z$ must be positive.
c) If I perform a linear regression of X − Y on Z. the $e^Z$ must be negative.
Thanks for your advice in advance.