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uncool

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Everything posted by uncool

  1. Gravity does not "always act[...] as an opposing force to inertia", which you'd know if you ever went skydiving. Gravity is a conservative force, and in a universe with just gravity, perpetual motion machines of the second kind are possible. The force that "opposes inertia" (more precisely, that equalizes velocities) is friction.
  2. You seem to be assuming only integer solutions are acceptable, which seems unwarranted. You can find a real a and b as long as x is at least 4 or negative.
  3. Because for any integer n, the corresponding subset of N will be finite. Which integer corresponds to the set of even integers?
  4. "What Max Planck did" included many things beyond defining a set of units.
  5. It isn't quantized, because what you've said isn't what "quantized" means.
  6. It doesn't. What values or states are restricted by choosing a unit system?
  7. Again, he didn't, because that's not what "quantized" means.
  8. "In 1898, Max Planck discovered that action is quantized, and published the result in a paper presented to the Prussian Academy of Sciences in May 1899.[24][25] At the end of the paper, Planck introduced, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as Planck's constant. Planck called the constant b in his paper, though h (or ħ) is now common. Planck underlined the universality of the new unit system [...]"
  9. Then it doesn't make sense to ask whether it is quantized; quantization happens to operators, or on a more meta level, classical theories.
  10. Because quantizing isn't as simple as "just multiplying it by G_p". If you mean whether it is used in a quantum theory of gravity, then yes, it is.The fact that it can be taken to be 1 is merely for convenience.
  11. ...whatever you mean by "it", the answer is no.
  12. ...no? The gravitational constant is defined by Newton's law of gravitation, F = G*m1*m2/r^2. Therefore, G = F*r^2/(m1*m2), so the units of G must be Force*distance^2/mass^2. Force = mass*acceleration = mass*distance/time^2, so G has units of distance^3/(mass*time^2).
  13. The statement of the theorem itself, as I have repeatedly shown. For that I apologize; the forum changed software a couple years ago in a really frustrating way, and I still haven't really tried to learn how the new LaTeX embedding works. Then ask about them. 1) That really is a bad reason for never having had to take it. 2) Either you misunderstood, or you were misinformed. 3) If you've never taken probability before, you should not be making pronouncements on it. 4) To be honest, you should learn the basics of probability theory before teaching it. Probability theory is a very well-developed field of math. The theorem you are trying to talk about is one of the most basic theorems in statistics - one of the subfields of probability - and one of the most necessary.
  14. "They" would not, as "they" have not in the past 118 years. You could try reading my post to find out. Or actually learning probability theory rather than making pronouncements about it. Choose any N, and some small epsilon. Then: For any epsilon and delta, there exists an N such that for n > N, Pr(|Y_n - p| > epsilon) < delta Equivalently, sum_{n*(p - epsilon) < i < n*(p + epsilon)} Pr((sum X_j) = i) > 1 - delta Equivalently, sum_{n*(p - epsilon) < i < n*(p + epsilon)} nCi * p^i * (1 - p)^j > 1 - delta. So take n large (say, larger than 100*(p - p^2)/epsilon^2), and find sum_{n*(p - epsilon) < i < n*(p + epsilon)} nCi * p^i * (1 - p)^j. I claim it will always be larger than 99%. Then explain what in them you disagree with, or think doesn't apply. What you've written shows no indication you even read what I wrote (especially since most of what I wrote are inequalities, not equations). Because, once again, it's about looking in a range. Every indication shows the problem is on the receiving end here, since you have failed to show that you even read what I wrote.
  15. 1) Mathematics has no Nobel Prize. 2) The proof has existed for centuries. As I have told you repeatedly. And as Ghideon has been more than willing to provide data for, and as I have been more than willing to provide data for. For any epsilon and delta, there exists an N such that for n > N, Pr(|Y_n - p| > epsilon) < delta Equivalently, sum_{n*(p - epsilon) < i < n*(p + epsilon)} Pr((sum X_j) = i) > 1 - delta
  16. In other words, there is literally nothing that could possibly convince you, and you aren't even going to try to understand the arguments we post. Do I have that right? I repeat: If you flip a coin 200 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 17%. If you flip a coin 2000 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 36%. If you flip a coin 20000 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 84%. This is an example where I literally chose the numbers (.495, .505, 200, 2000, 20000) arbitrarily (because they were easy for me). The probability of being close to 1/2 approaches 1 - for any reasonable choice of "close".
  17. As with any proof, we should start with the statement we are trying to prove, and then start the proof proper. The statement of the weak law of large numbers for a binary variable is the following: Let X_1, X_2, ..., X_n be independent, identically distributed binary variables (in layman's terms: they're coinflips that don't affect each other and have the same probability p). Define Y_n = (X_1 + X_2 + ... + X_n)/n. Then for any epsilon > 0, lim_{n -> infinity} Pr(|Y_n - p| > epsilon) = 0. Writing out the limit: for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|Y_n - p| > epsilon) < delta To prove it, we will need a few lemmas. Definition: X and Y are independent if for any outcomes i, j, P(X = i, Y = j) = P(X = i) * P(Y = j). Definition: For a discrete variable X, E(X) = sum_i i*P(X = i) Note the summation in the above. Lemma 1: For any two independent variables X and Y, E(XY) = E(X) E(Y). Proof: E(XY) = sum_{i, j} i*j*P(X = i, Y = j) = sum_{i, j} i*P(x = i) * j * P(Y = j) = (sum_i i*P(x = i)) (sum_j j*P(x = j)) = E(X) E(Y) Lemma 2: Assume X is a variable with all positive outcomes. Then for any a, P(X > a) <= E(X)/a. Proof: E(X) = sum_i i*P(X = i) = sum_{i > a} i*P(X = i) + sum_{i <= a} i*P(X = i) >= sum_{i > a} a*P(X = i) + sum_{i <= a} 0*P(X = i) = a*sum_{i > a} P(X = i) = a*P(X > a), so P(X > a) <= E(X)/a. Lemma 3: If X and Y are independent, then X - a and Y - b are independent. Left to the reader. Lemma 4: E(X - p) = 0. Left to the reader. Lemma 5: E((X - p)^2) = p - p^2. Left to the reader. Lemma 6: For any variables X and Y, E(X + Y) = E(X) + E(Y) (no assumption of independence needed). Left to the reader. Now, as is usual for limit proofs, we work backwards from the statement we want to prove to the statements we can prove. We want to prove that for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|Y_n - p| > epsilon) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|(sum(X_i))/n - p| > epsilon) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|sum(X_i) - p*n| > epsilon*n) < delta I want to note here, once again, that this shows what I've been saying: that this is about a range of possibilities. In this case, that range is epsilon*n around the "perfect" outcome. Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr((sum(X_i) - p*n)^2 > epsilon^2*n^2) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr((sum(X_i - p))^2 > epsilon^2*n^2) < delta Applying lemma 2 (since squares are always positive), we know this is true as long as E((sum(X_i - p))^2) < delta*epsilon^2*n^2, because then Pr((sum(X_i - p))^2 > epsilon^2*n^2) <= E((sum(X_i - p))^2)/(epsilon^2 * n^2) < delta. (sum(X_i - p))^2 = sum_{i, j} (X_i - p)(X_j - p) = sum_i (X_i - p)^2 + sum_{i =/= j} (X_i - p)(X_j - p), so E((sum(X_i - p))^2) = E(sum_i (X_i - p)^2 + sum_{i =/= j} (X_i - p)(X_j - p)) By lemma 6, we can split this sum up into individual terms. The first term is sum_i E((X_i - p)^2) = sum_i (p - p^2) = n*(p - p^2) by lemma 5. The second term is sum_{i =/= j} E((X_i - p)(X_j - p)) = sum_{i =/= j} E(X_i - p) E(X_j - p) by lemma 1, = 0 by lemma 4. So the condition we want is n*(p - p^2) < n^2*delta*epsilon^2, or n > (p - p^2)/(delta*epsilon^2). Which means choose N = ceil((p - p^2)/(delta*epsilon^2)), and the statement follows. This proof generalizes quite easily; all that's necessary is to replace p by E(X_i) and (p - p^2) by E((X_i - E(X_i))^2). I was waiting for you to show any interest in the actual proof, rather than insisting that it hadn't been proven.
  18. I am writing the following in large font at the beginning and end of the post because it is an offer you have repeatedly ignored, and it is likely central to your confusion. I am offering to prove the weak law of large numbers for a binary variable (i.e. a biased or unbiased coin) using "summing probabilities", i.e. the method I have been using this entire thread. Do you accept? Then why are you rejecting the summation? Why not, when the law of large numbers isn't about the single possible outcome? I don't know where you are pulling this bullshit from, but it is bullshit. That is exactly the point I am saying is irrelevant. This isn't about "benefits and problems". It's about which method is correct. And I guarantee that summation is correct, by the laws of probability. I am offering to prove the weak law of large numbers for a binary variable (i.e. a biased or unbiased coin) using "summing probabilities", i.e. the method I have been using this entire thread. Do you accept?
  19. And why should we want area under a curve for this specific application? Why is that relevant to this particular calculation? The question isn't about whether one application is "better" than the other. The question is which one is correct. And the laws of probability explicitly say that summing is correct. If you mean that someone who thinks "adding together probabilities" (I assume you mean the method I have been demonstrating) can't get the result of the law of large numbers, then you are wrong.
  20. Because an integral is a particular limit that is not being taken here. Further, an integral is a limit of summation, not the other way around. I have offered to provide the proof of the law of large numbers in this thread multiple times.
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