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-   Chapter 1 - The Learning Problem (http://book.caltech.edu/bookforum/forumdisplay.php?f=108)

 ivanku 04-18-2012 12:59 AM

Quote:
 Originally Posted by yaser (Post 1288) This is approximately since . Numerically, .
Is it correct to assume that from the coins tossing simulation in homework 2 assignment one should be close to that limit (where we're tossing 1000 coins 10 times and is the fraction of heads obtained for the coin which had the minimum frequency of heads)?

 elkka 04-18-2012 03:49 AM

The law of big numbers states that the average $\nu_min$ is close to the $E{\nu_min}$.

$E\nu_min$ can be calculated directly for this experiment.
$P(\nu_min=0)$=0.623576
$P(\nu_min = 0.1)$ = 0.3764034
$P(\nu_min = 0.2)$ = 0.00002;
and $P(\nu_min>=0.3)=0$ for the purposes of calculating the mean.

Therefore, $E(\nu_min)$=0.037644, and the average proportion of heads for c_min should be close to this number.

 SamK52 04-18-2012 06:43 AM

Allow me to format your post:

Quote:
 The law of big numbers states that the average is close to the . can be calculated directly for this experiment. and for the purposes of calculating the mean. Therefore, , and the average proportion of heads for should be close to this number.

 elkka 04-18-2012 11:26 AM

Quote:
 Originally Posted by SamK52 (Post 1399) Allow me to format your post:

 yaser 04-18-2012 01:57 PM

Quote:
 Originally Posted by elkka (Post 1413) Please, allow me to ask how you did it?
http://book.caltech.edu/bookforum/sh...77&postcount=1

 rohanag 04-18-2012 01:58 PM

how did you calculate those probability values? ( nu_min = 0, 0.1, 0.2 )

 elkka 04-19-2012 04:10 AM

Thank you, Professor.

This how I calculate the probabilities. Let - the number of heads for , and let be the number of heads in i-th experiment (out of 1000). Then, as Professor has shown previously,

Now, . Therefore,

Next,

Next,

The rest can be calculated directly too, but they are essenctially 0 for the purpose of calculating the mean.

 rohanag 04-19-2012 08:44 AM

thank you for the detailed explanation.

 nyxee 07-13-2012 05:55 PM

the answer is very clear but how do we know when to use this not.

to clarify my question, if say P(ten heads)=p and P(not ten heads)=q (=1-p). why does using (p^1000) give the wrong answer?

 htlin 07-15-2012 04:56 AM