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-   -   Homework Q5 (http://book.caltech.edu/bookforum/showthread.php?t=4590)

Andrew87 02-21-2015 10:34 AM

Homework Q5
 
Hi, I'm working on the 6-th question but I can't get the solution.

I solved Q5 and I found a probability to get \upsilon = 0 on just one sample (N=10) of 0.0047. Then to compute the probability to get at least one sample with \upsilon = 0, I modeled the number of samples with 0 red marbles N_0 with a binomial random variable B(0.0047, 1000). Hence, the probability I need is

P(N_0 \geq 1) = 1 - P(N_0 = 0)

and

P(N_0 = 0) = (1 - 0.0047)^{1000} = 0.009

Finally P(N_0 \geq 1) = 0.991, so my closest answer is [3] = 0.550, however the solution report [c] as right answer. Where did I make a mistake ? Thank you in advance for your time (and sorry for my english).

yaser 02-28-2015 07:55 PM

Re: Homework Q5
 
Quote:

Originally Posted by Andrew87 (Post 11928)
Hi, I'm working on the 6-th question but I can't get the solution.

I solved Q5 and I found a probability to get \upsilon = 0 on just one sample (N=10) of 0.0047. Then to compute the probability to get at least one sample with \upsilon = 0, I modeled the number of samples with 0 red marbles N_0 with a binomial random variable B(0.0047, 1000). Hence, the probability I need is

P(N_0 \geq 1) = 1 - P(N_0 = 0)

and

P(N_0 = 0) = (1 - 0.0047)^{1000} = 0.009

Finally P(N_0 \geq 1) = 0.991, so my closest answer is [3] = 0.550, however the solution report [c] as right answer. Where did I make a mistake ? Thank you in advance for your time (and sorry for my english).

I did the math and the probability that one sample has no red marbles, which is 0.45^{10} evaluates to approximately 0.00034 rather than 0.0047.

Andrew87 03-03-2015 07:25 AM

Re: Homework Q5
 
Thank you very much Prof. Yaser !

I did a mistake in calculating the probability of the Homework 4. This was caused by the fact that I used the Hoeffding inequality to compute this probability instead of a simple binomial distribution. Now, I'm wondering a more general question: is this slack between the exact probability one gets with the binomial (in this case) and the bound provided by the Hoeffding inequality typical ? Thank you very much in advance for your answer.

yaser 03-03-2015 08:47 PM

Re: Homework Q5
 
Yes. Since the bound is the same for cases that have different probabilities, it is forced to be loose in some of the cases.

Andrew87 03-04-2015 04:15 AM

Re: Homework Q5
 
Thank you very much for your answer Prof. Yaser !


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