For a 2-dimensional random vector, express the probability of being in a rectangle in terms of CDF

Question

For $a_1\le b_1$ and $a_2\le b_2$, show that $$P\{ a_1<X_1\le b_1,a_2<X_2\le b_2 \}= F(a_1,a_2)+F(b_1,b_2)-F(b_1,a_2)-F(a_1,b_2)$$ Here $F(a,b)=P\{X_1\le a,\ X_2\le b\}$, the cumulative distribution function.

I know that I need to write down the probability of rectangle as a combination of four probabilities which then I would write down as distribution functions of F. But I don't know how.

Answer 1

Figured out myself: $P\{ a_1<X_1\le b_1,a_2<X_2\le b_2 \}$ can be written as $$P(X_1\le b_1,X_2\le b_2)+P(X_1\le a_1,X_2\le a_2)-P(X_1\le a_1,X_2\le b_2)-P(X_1\le b_1,X_2\le a_2)$$ which is equal to $$ F(a_1,a_2)+F(b_1,b_2)−F(b_1,a_2)−F(a_1,b_2)$$ -- Justin