I am studying for an exam and came across this example to calculate probability $P(X_1\geq1/2, X_2\geq1/2)$ for $F_{X_1,X_2}$ = $1/2x_1x_2(x_2^2+x_1^2)$ if $0\leq x_1\leq 1,0\leq x_2\leq 1$.
The provided solution states:
$P(X_1\geq1/2, X_2\geq1/2)$ = 1 - (F(1/2,1) + F(1,1/2) - F(1/2,1/2)) [brackets added]
Can someone provide a detailed explanation of this solution? I cannot follow the steps provided in this solution.
I believe the last two signs should be switched. Please double check.
Then it can be thought of as this. Picture the support which is the unit rectangle. Divide up the rectangle into four equal sub-pieces. The probability desired is the volume over the top right corner. It can be obtained as the volume over all pieces (which is 1), taken away the volume over the left two pieces $F(0.5, 1)$, then take away the volume over the bottom two pieces $F(1, 0.5)$, and then adding back up the volume over the bottom left corner $F(0.5, 0.5)$.
In a way, it is like the inclusion-exclusion principle.