I Confusion over using integration to find probability

[Of Mike and Men]

Hey everyone, first, let me say I understand the complement rule. Where I am confused is over the integration. My professor said that suppose you have a continuous cumulative distribution function F(x) = 1-e^-x/10, if x > 0 (0, otherwise). And suppose you want to find P(X>12) you can use the complement rule 1-P(X<=12). Which is equivalent to 1-F(12) [note he said this works for all cases, not just this example].

My question is why isn't it 1-[F(12) - F(0)]?

This is really tripping me up. If your x can take all probabilities from 0 to 12, don't you want to find the area from 0 to 12 and not just F(12)?

I know this is a method of simplifying the integral since you have an improper integral and have to evaluate a limit (supposing you don't use the compliment rule). But why does this work for all cases?

 

[Orodruin]

What value does F(0) take?

Edit: Also, the cdf F(x) is the probability of taking any value less than or equal to x. By definition, you therefore have P(X ≤ 12) = F(12).

 

[Of Mike and Men]

What value does F(0) take?

In this case 1 - 1 = 0. Meaning that you have 1 - [F(12) - 0], in this case. But he said it is true for all cases...

 

[Orodruin]

In this case 1 - 1 = 0. Meaning that you have 1 - F(12) - 0, in this case. But he said it is true for all cases...

Read my edit above.

 

[Of Mike and Men]

Read my edit above.

I guess, why is this the definition? Perhaps this is more calculus related and I'm not remembering. It seems vaguely familiar. I guess it relatively makes sense since F(A) - F(B) would be the area between the two points. Then F(A) would be the entire area up until A since you'd have no lower bound.

 

[Orodruin]

I guess, why is this the definition?

Like most defined things, it is defined because it is useful.

Indeed, F(B)-F(A) would be equal to P(A<X≤B).