Who can help me to interpret the following expression?

Question

I don't understand the second line of the following expression. Why does he use conditional expectation? and can you explain the following calculation process to me? Thanks.

Answer 1

Do you agree with the following(?): \begin{align} \text{Pr}\left( (X_{1}^{\star} - X_{2}^{\star}) (Y_{1}^{\star} - Y_{3}^{\star})>0 \right) &= \sum_{X_{1}^{\star}, Y_{1}^{\star}} \text{Pr}\left( (X_{1}^{\star} - X_{2}^{\star}) (Y_{1}^{\star} - Y_{3}^{\star})>0\; \vert\; X_{1}^{\star}, Y_{1}^{\star} \right) \cdot \text{Pr}\left( X_{1}^{\star}, Y_{1}^{\star} \right)\\ &= \text{E}_{X_{1}^{\star}, Y_{1}^{\star}} \left[\text{Pr}\left( (X_{1}^{\star} - X_{2}^{\star}) (Y_{1}^{\star} - Y_{3}^{\star})>0\; \vert\; X_{1}^{\star}, Y_{1}^{\star} \right)\right] \end{align} To verify that, you may want to check the "Law of total probability".

For the next step, you have to think the following: When does it hold that $(X_{1}^{\star} - X_{2}^{\star}) (Y_{1}^{\star} - Y_{3}^{\star})>0$? There are two cases:

• Either $Y_{3}^{\star} \le Y_{1}^{\star}$ and $X_{2}^{\star} \le X_{1}^{\star}$ simultaneously, or
• Either $Y_{3}^{\star} > Y_{1}^{\star}$ and $X_{2}^{\star} > X_{1}^{\star}$ simultaneously.

Recall that $X_{1}^{\star}$ and $Y_{1}^{\star}$ are now constants (given). We also know that $X_{i}^{\star}$ and $Y_{i}^{\star}$ have marginals that are uniform between $0$ and $1$. Now, I suspect that $X_{i}^{\star}$ and $X_{j}^{\star}, Y_{k}^{\star}$ for $i \neq k, i\neq j$ are independent (because they have a different subscript - am I wrong?) If I am correct, then taking into account the fact that $X_{i}^{\star}$ is uniformly distributed on $(0,1)$, we have $$ \text{Pr}(X_{2}^{\star} > X_{1}^{\star}) = (1 - X_{1}^{\star}), \quad \text{and} \quad \text{Pr}(X_{2}^{\star} \le X_{1}^{\star}) = X_{1}^{\star}. $$ Similarly, for $Y_{3}^{\star}$.

Then, the third step follows from the linearity of expectation and the next by the definition of covariance. Note that the expected value of both $X_{i}^{\star}$ and $Y_{i}^{\star}$ is equal to $1/2$.