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You are not logged in. #1 20130124 01:37:08
The new dice pdf is out!Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #2 20130217 06:27:56
Re: The new dice pdf is out!Hi bobbym My answer to the last question in problem #10 regarding the case in which we want a number to appear 3 times before stopping: First, we need to find the probability P(X=i) that the first number that appears 3 times is in the ith postition: For the first number that appears 3 times to be in the ith position, it must also appear exactly twice in the previous i1 places, and all other number can appear at most 2 time. This means, also, that, by the Pigeon hole principle, i can take values between 3 and 13, only. Now, the number of ways we can choose two numbers from the first i1 is , and the probability that they are the same as the ith number is . The rest of the numbers can appear at most 2 times, so we can use the GF to model them. Because the total number of other numbers in the first i1 is i3, and because there are only 5 possible choices for those i3 numbers, we will model this with the GF , whose x^(i3) term's coefficient is equal to the number of ways of distributing the i3 numbers among the five possibilities. But this only gives us the possible distribution of those i3 numbers, without caring for theor order in the throw sequence. To account for that we need a factor of (i3)!. So, finally, the number of ways the i3 numbers can be distributed is . Also, the probability that those numbers will have the values assigned to them is Combining this with the previous results gives us the probability of . To get the expected value, we use the standar formula E(X)=∑ i*P(X=i): . Now we can use a CAS to find the value of this sum, or to get a closed form. The following code in Maxima does that: Code:sum(i*(i3)!/(6^(i1))*binomial(i1,2)*coeff(ratexpand((1+x+x^2/2)^5),x^(i3)),i,3,13); 4084571/559872 Which gives us the final answer of .The result can be generalized from 3 to k appearances before stopping as follows: In Maxima, we could implement this as: Code:f(k):=sum(i*(ik)!/(6^(i1))*binomial(i1,k1)*coeff(ratexpand((sum(x^j/j!,j,0,k1))^5),x^(ik)),i,k,6*k5); Last edited by anonimnystefy (20130321 07:45:50) The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #3 20130217 08:41:05
Re: The new dice pdf is out!Hi anonimnystefy; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #4 20130217 08:49:52
Re: The new dice pdf is out!Thanks! The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #5 20130217 09:22:52
Re: The new dice pdf is out!I have a little better understanding of it now and it has worked for all the examples given and for a few simulations of numbers that are bigger than 2 and 3. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #6 20130217 09:25:36
Re: The new dice pdf is out!That is great! I will wait for him to answer my email now. Last edited by anonimnystefy (20130217 09:25:44) The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #7 20130217 09:32:12
Re: The new dice pdf is out!There are one or two spots that might be simplified or maybe not. I do not yet know about an asymptotic form. It would interesting to be able to answer at least approximately for say 1000000 in a row. Last edited by bobbym (20130218 00:06:48) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #8 20130217 10:11:28
Re: The new dice pdf is out!It seems that the inner sum has a closed form in terms of the Gamma and the upper incomplete Gamma function. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #9 20130217 10:12:29
Re: The new dice pdf is out!Yes, there is a closed form for much of that. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #10 20130217 11:24:06
Re: The new dice pdf is out!That sum is troubling me. Is there anyway to get rid of it? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #11 20130217 11:39:53
Re: The new dice pdf is out!Which sum are we talking about? In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #12 20130217 11:49:03
Re: The new dice pdf is out!The inner sum. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #13 20130217 12:45:03
Re: The new dice pdf is out!That is for the gf, you do not have to use that approach. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #14 20130217 12:47:14
Re: The new dice pdf is out!I don't understand you. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #15 20130217 12:54:03
Re: The new dice pdf is out!There was another way to generate T[n,k], it used a series. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #16 20130217 12:59:50
Re: The new dice pdf is out!Yes, but it cannot be used for the generalized case. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #17 20130217 13:05:40
Re: The new dice pdf is out!Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #18 20130217 20:28:59
Re: The new dice pdf is out!I will if he says he's interested.in seeing it. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #19 20130217 21:56:19
Re: The new dice pdf is out!Hmmm, he did ask for your method. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #20 20130217 22:58:52
Re: The new dice pdf is out!Yes. He asked for the method, but not the whole solution. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #21 20130217 23:04:49
Re: The new dice pdf is out!Hi; Last edited by bobbym (20130217 23:14:32) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #22 20130217 23:19:37
Re: The new dice pdf is out!Lie about this:
The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #23 20130217 23:21:57
Re: The new dice pdf is out!Could be that I have just got up or your solution or both but I am not particularly sharp right now. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #24 20130217 23:23:13
Re: The new dice pdf is out!Maybe it's because of your back. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #25 20130217 23:24:11
Re: The new dice pdf is out!I have been working hard on it but unfortunately no. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 