I have a continuous random variable Y with its mean conditioned on another random (discrete) variable X as follows.
m0 = E(Y|X=0) = 12 m1 = 3 m2 = 7
and I want to find mY|X<2 = E(Y|X<2). Is this intuitive. Because supposedly it is not and my immediate reaction is to say it is simply the E(Y|X = 0) + E(Y|X = 1)
Thanks in advance
EDIT: I do have some information on the variable X but it seemed to not be helpful to me.
p0 = P(X = 0) = 0.2 p1 = 0.7 p2 = 0.1
The conditional expectation of $Y$ given an event $A$ is $$E[Y \mid A] = \frac{E[Y \cdot 1_A]}{P(A)},$$ where $1_A$ is an indicator random variable that equals $1$ when the event $A$ occurs, and equals zero otherwise.
In your case you have $A = \{X < 2\}$. The denominator is $P(A) = P(X=0) + P(X=1)$. By the law of total expectation, the numerator is $$E[Y \cdot 1_A] = \sum_{x=0}^2 E[Y \cdot 1_A \mid X=x] P(X=x) = E[Y \mid X=0] P(X=0) + E[Y \mid X=1] P(X=1).$$