Let's assume that X is an image, Y is the identity and C is the count. More specifically, $\text{P}(\text{Y}=n|\text{X}=m)$ denote the probability of position $m$ belonging to the identity n, and $\text{P}(\text{C}=k|\text{X}=m)$ denote the probability of position $m$ having $k$ identities. Note that, Y and C are conditional indenpent on X, i.e., $\text{P}(\text{Y,C}|\text{X})=\text{P}(\text{Y}|\text{X})\text{P}(\text{C}|\text{X})$.
Now, I want to calculate the expectation of the count of the identity $n$ at position $m$, which can be written as: $$\text{P}(\text{Y}=n|\text{X=m})(\sum_{k}\text{P}(\text{C}=k| \text{X}=m)\cdot k)$$ where $\sum_{k}\text{P}(\text{C}=k| \text{X}=m)\cdot k$ calculates the expection of count at potion $m$, and $\text{P}(\text{Y}=n|\text{X=m})$ assigns the count to the identity $n$.
My question is, is this expression can be written into any expectation formulations? Such as $\text{E}(\text{C},\text{Y}=n|\text{X}=m)$?