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"the probability that N is a perfect n^nth power..." Can someone explain what probability means here in relation to N? From my understanding, it depends on wha
by ambrop7 10y ago
"the probability that N is a perfect n^nth power..."
Can someone explain what probability means here in relation to N? From my understanding, it depends on what N is for you. If it's a constant, that probability is obviously 0 or 1. So that can't be it. Then N must be some kind of random variable. But with what distribution? And in what kind of system can the probability of event(random_variable) involve random_variable itself?
- lsh123 10y agoThe meaning here is: pick a positive integer N, what is the chance (aka probability) of it being an n-th power of another positive integer. And you are correct, this probability depends on N.
- ambrop7 10y agoI don't get it. The probability must depend on how likely I am to select any specific integer, i.e. probability mass function of N. The probability cannot depend on the value of the random variable itself but could involve any parameters that define its distribution. I would consider something like the following a valid question: "Let N be a random integer between 0 and M-1 with uniform distribution. What is the probability that N is even?" Then an answer could be "the probability that N is even is 1/2 if M is even and (M+1)/(2M) if M is odd". See this does not involve N but does involve M which is a parameter for the distribution of N. Your explanation seems to invoke some "common sense" which I am not able to unify with my understanding of probability theory.
- sparky_z 10y agoIntuitively, you can think of N as a rough measure of the "ballpark" around where you are in the number line. As you move up to higher numbers, perfect powers get more and more scarce, so the probability of finding one by picking a number at random becomes lower.
- ikeboy 10y agoOne meaning is that summing the probabilities for N=1,2,3, ... and checking how many numbers actually meet the criteria yield the same result in the limit. More precisely, their ratio goes to 1 as N goes to infinity. Edit: define f(N) as the number of numbers below N that match. Consider the function f(N)/N. We call a function of N the probability of N matching if it asymptomatically approaches f(N)/N.
- euyyn 10y agoI think you should just think of it as the density and that's it.
- GFK_of_xmaspast 10y agoHere are a couple refs: https://www.amazon.com/Introduction-Analytic-Probabilistic-Cambridge-Mathematics/dp/0521412617 https://www.amazon.com/Introduction-Analytic-Probabilistic-C... http://algo.inria.fr/seminars/sem96-97/deshouillers.pdf http://algo.inria.fr/seminars/sem96-97/deshouillers.pdf
- lsh123 10y agoImagine you don't have an easy way to solve the equation x^n = N where x is an integer. Yet, for a given N you want to find a probability that such x (solution) exists. There is nothing here about how you select N.
- spott 10y agoYou are thinking "given N, what is the probability that N is a perfect nth power?". Think about it in the reverse. "given N, what is the probability that the nth root of N is an integer". For a whole bunch of different Ns, we will get some real number between 0 and 1. To match up to your example. "For a number M, m is the remainder of M: m = M - floor(M), what is the probability that m is zero given M"
- gohrt 10y agoIt's a simple density in the integers. Either globally: Proposition P is true for n of the numbers from 1:N -> p(P,N) = n/N or locally: Proposition P is true for n of the numbers from N-k to N+k -> p(P, N, k) = n/k and pick a reasonable k (where p(P, N, K) is relatively stable for a neighborhood around k)