Expection of discrete random variables (dependent)

Question

Let $X$ and $Y$ be two discrete random variables.

We know $E(XY)=\sum \sum X_{i}Y_{j}P(X_{i},Y_{j})$

Now suppose $X$ represents income and $Y$ consumption. In my observations, every time I ask a person's income, I also ask for his consumption.

This means that $P(X_{i},Y_{j})=0$ for $i\ne j$

Then we can rewrite $E(XY)=\sum X_{i}Y_{i}P(X_{i},Y_{i})$

Is there any specific name for these kind of cases? I think saying that $X$ and $Y$ are dependent isn't enough for $E(XY)$ to be equal to the above.

Edit: I'm thinking maybe (dependent variables $\cap$ uniformly distributed variables) gives us the above case.

Answer 1

Assuming $\{X_i\},\{Y_j\}$ are the support of $X$, $Y$, it means that the variables are fully determined by each other.

$$\mathsf P(X=X_i) ~=~ \sum_j \mathsf P(X=X_i, Y=Y_j) ~=~ \mathsf P(X=X_i, Y=Y_i) \\\therefore \mathsf P(Y=Y_i\mid X=X_i)=1$$