P(a1<X<a2, b1<Y<b2) Understanding

[silvermane]

Homework Statement

My professor showed us the identity,
P(a1<X<a2, b1<Y<b2) = F(a1,b1)+F(a2,b2)-F(a1,b2)-F(a2,b1)
where (X,Y) are jointly distributed rvs with a joint cdf of F(x,y) = P(X$$\leq$$x, Y$$\leq$$y) and a1<a2, b1<b2.
It's not homework that we turn in, but a supplement that she showed us to think about and look at.

The Attempt at a Solution

I understand what it is geometrically with integration, however I want to see how it could be proved using set theory and probability identities such as
P(AUB) = P(A) +P(B) - P(AB), etc.

Thanks so much for your help in advance!

 

[mighty2000]

Thank you for sharing your professor's identity and your attempt at understanding it. It is always great to see students taking the initiative to understand concepts beyond what is required for homework.

To prove this identity using set theory and probability identities, we can start by defining the events A = {a1<X<a2} and B = {b1<Y<b2}. Then, we can rewrite the given identity as:

P(A∩B) = P(A) + P(B) - P(A∪B)

Using the definition of joint cumulative distribution function (cdf), we can rewrite P(A∩B) as F(a2,b2) - F(a1,b2) - F(a2,b1) + F(a1,b1). Similarly, P(A) and P(B) can be written as F(a2,∞) - F(a1,∞) and F(∞,b2) - F(∞,b1), respectively. Finally, P(A∪B) can be written as F(a2,∞) - F(a1,∞) - F(∞,b1) + F(a1,b1).

Substituting these values into the rewritten identity, we get:

F(a2,b2) - F(a1,b2) - F(a2,b1) + F(a1,b1) = F(a2,∞) - F(a1,∞) + F(∞,b2) - F(∞,b1) - F(a2,∞) + F(a1,∞) + F(∞,b1) - F(a1,b1)

Simplifying, we get:

F(a2,b2) = F(a2,∞) + F(∞,b2) - F(∞,b1) - F(a1,∞) + F(a1,b1)

which is the same as the given identity.

I hope this helps in understanding the proof of this identity. Keep up the good work in exploring and understanding concepts beyond what is required. That is what makes a great scientist.