Is my understanding of a multivariate discrete conditional expectation calculation correct?

Say I have three discrete variables $A$, $B$ and $C$ that all have the range $\{-3, -2, -1, 0, 1, 2, 3\}$.

The domain of my random variable is formed by the Cartesian product $A \times B \times C$ so the space looks like $\{(-3,-3,-3),(),...\}$. Each variate, say $(-3,-3,-3)$ has a probability associated with it, forming the discrete probability distribution $p()$ of the random variable.

I would like to obtain a conditional expectation $E(A \times B \times C \, | \,a = 1)$.

As far as I understand this would be:

$$E(A \times B \times C \, | \,a = 1) = \bigg(1, \sum_{B \times C}\bigg( b \cdot p\big((a = 1,b,c)\big) \bigg), \sum_{B \times C}\bigg( c \cdot p\big((a = 1,b,c)\big) \bigg)\bigg)$$

Is this correct? Sorry if the notation is a little convoluted, I have tried to keep my intention unambiguous.

• I'm not sure, but don't you need to have the word independent somewhere in there? – Ovi Sep 7 '17 at 12:33
• What does $a$ have to do with $A$, $B$ and $C$? – Therkel Sep 11 '17 at 20:20