Advanced Probability (math or stat majors probably only need apply)

MagicFlea · #1 02-13-2006, 02:43 PM

I want help on my homework [img]/images/graemlins/smile.gif[/img]

I need to show a certain probability converges >0 as the limit goes to inf.

Pn(i,i)= (1-e^-(n^c)) for arbitrary small constant c

Need to show: PI (n=1 to inf) (Pn(i,i)) > 0

I've simplified it to showing (1-e^-(n^c))^n does not dissapear as n->inf which should be provable with the binomial expansion but its not working for me. If you have an idea on either front I'd be happy [img]/images/graemlins/smile.gif[/img]

(this relates to cooling schedules for annealating processes, if it interests you)

AaronBrown · #2 02-13-2006, 11:33 PM

There's something wrong with the problem, or perhaps in trying to type it in text. Can you type it in LaTex?

I assume Pn is supposed to be P-subscript-n; but what are the arguments (i,i)? They don't appear in the expression. By PI I assume you mean product. But what goes to infinity? Do you mean the limit of the sequence of products from n=1 to m as m goes to infinity?

Siegmund · #3 02-14-2006, 05:07 AM

I'm sure you know we can't give you a proof, just point you in a direction that might be fruitful......

the product of the P_i is zero iff the sum of the logs of the P_i goes to -infinity.

Can you say anything about this sum?

There are some handy bounds above and below on the value of log(1+x) for sufficiently small x.

SunOfBeach · #4 02-15-2006, 01:25 PM

I'm having a bit of a problem with your notation/description. Could you rewrite this for me, and/or pm me? i'd be happy to help you through it...

BillC · #5 02-16-2006, 04:20 PM

I agree with the others that you should be more clear (I would take points off for sloppiness). I presume 0<c<1.

It is unethical to get 2+2'ers to do your HW for you...Otoh, you can't believe everthing you read here, so I'll give a hint: Fall off a log and go to the hospital.