Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Hey I tried searching online but I can't find this proof based on the CDF.

$$P(a < X \le b) = F(b) - F(a)$$

Thanks for any help.

share|improve this question
1  
I think it should be clear if you write out what $F(b)$ and $F(a)$ are. –  pedrosuavo Apr 28 '13 at 4:48
1  
I mean, there isn't really much to prove. –  pedrosuavo Apr 28 '13 at 4:48
    
@wzsun Which parts of the accepted answer were you unaware of when you asked the question? –  Did Apr 28 '13 at 6:47
    
I must admit, I have always found probability to feel a bit unnatural.... it requires a lot of different thinking. And for me, I was confused at the start of reading Rosenthal because PDF is just called distribution function. –  Squirtle Apr 29 '13 at 5:13
add comment

1 Answer

up vote 1 down vote accepted

By definition the CDF (cumulative density/distribution function) is computed by taking a "cumulative" integral up to the given point of the PDF $(f_X(x)):$ $$F(a):=P(X \le a):=\int_{-\infty}^a f_X(x)dx=1-P(a<X)=1-\int_{a}^\infty f_X(x) dx$$

I included the part towards the right because I thought it might help you 'see' what's going on.

Likewise $$P(X\le b):=\int_{-\infty}^b f_X(x)dx=F(b)$$

So $$P(\{a\le X \} \cap \{X\le b\})=P(a\le X \le b) = \int_{-\infty}^b f_X(x)dx - \int_{-\infty}^af_X(x)dx=F(b)-F(a)$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.