Calculate expectation of a function with two dependent random variables Hi Cross Validated community,
My question has to do regarding expectation of a multiplication of two random variables that are dependent.
Assume there are two random variables, one discrete: $G \in \{0,1\}$ and one continuous: $D \in (0, 1000)$, the two variables are dependent $G \rightarrow D$. The payoff function $F$ is then defined as follows:
$$
F(g, d) = \begin{cases}
    1.5 \cdot d & \text{if } g = 1 \\
    -10 & \text{if } g = 0
\end{cases}
$$
How can one calculate the expectation
$$
\DeclareMathOperator{\E}{\mathbb{E}}
\mathop{\E}_{P(G,D)}[F(g,d)]$$
given that the joint probability $P(G,D)$ is not known and difficult to model. Currently I have $E[D]$ out of an estimator and $P(G)$ out of a classifier.
If I assume independence between $G$ and $D$, the expression should be straightforward:
$$
\E[F(g,d)] = P(G=0)\cdot (-10) + P(G=1) \cdot \E[D] \cdot 1.5
$$
Now the question is how to do this for G and D being dependent. What would be the best way to go about correctly expressing and calculating this expectation? Some of the ideas are:

*

*Perhaps I may somehow use the following trick by adding covariance somehow, (It is however unclear to me how can I correctly add this into the expectation formula):

$$
\E[GD] = \E[G] \cdot \E[D]+cov[G,D]
$$


*Or perhaps I will have to model $P(D|G)$ to calculate the expectation?

Thank you for your help and ideas. : )

If you assume dependence, the expression is given by:
$$\mathbb E[F(g,d)] = P(G=0)\cdot (-10) + P(G=1) \cdot \mathbb E[D|G=1] \cdot 1.5$$
Can you estimate $\mathbb E[D|G=1]$ ?
