Let be the CDF of the standard normal density function. I want to evaluate:
. Clearly this is equal to (we have that a > 0).
What I would like to know is whether I can bring the limit inside. The CDF is right-continuous and so I would usually have no issues with bringing it inside. However is defined? If it is NOT defined then since is itself equal to , it seems that I'm not allowed to bring the limit inside of in this instance? But if it is defined at +infinity then I can bring the limit inside.
Last edited by derksheng; 10-15-2012 at 10:37 PM.
Lol. Sorry. Of course it's 1. been typing too much latex in last 5 hours.
And as a side note you need to have more than just "right continuous" to be able to bring a limit inside the integral.
I don't have emotions and sometimes that makes me very sad.
derksheng (10-15-2012)
I am confident that we can extend it to because we would just be taking the integral of the PDF from to which is well defined as equal to 1.
However I'm less confident about fixing x = in N(x), because I haven't done the math to show that we can integrate from to and get 0 without any problems.
It doesn't help that these notes make the claim that the domain is :
http://ocw.mit.edu/courses/electrica...JF08_lec05.pdf
derksheng (10-15-2012)
Okay good so the domain is the extended reals! Great!!!
Last edited by derksheng; 10-16-2012 at 02:40 AM.
To finalize this thread, N(.) is both right and left continuous so it is always fine to bring in the limit.
But in a measure theoretic sense you are bringing it inside the limit because what is happening is
Further note that continuity isn't all you need to bring limits inside!
Let
Now is continuous for all n. Also for any value of x we have as . However
for all n.
So in this case
If there is one thing I learned in measure theory it's that you need to be careful when moving limits around.
I don't have emotions and sometimes that makes me very sad.
derksheng (10-18-2012)
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