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Suppose the integers in the array are distinct. Therefore, only one permutation of the integers corresponds to a sorted array. Moreover, suppose you use the Fisher-Yates shuffle (or anything where each permutation is equally likely). Since there are $N!$ permutations, each permutation has probability $1/N!$ of appearing. Now, the expected number of trials ...

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The statement about the digits of $\pi$ is not known to be true. But the corresponding statement is true for a random sequence of digits. Say we're looking for the sequence 1243. That could be the first four digits, or the next four digits... Probability problems with "and" are often easier than problems with "or". Say the sequence 1243 does not appear in ...

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For the first question : An irrational number $x$ is called a 'normal number' if for any natural number $n$ with $m$ digits, the density of $n$ in the decimal digits of $x$ is $\frac{1}{10^m - 10^{m-1}}$ in all bases (i.e. you see every natural number (with the same number of digits) in the decimal digits the same time). It is a conjecture that whether $\... 0 If you are only concerned about the frequency, then a simple computer program which goes through all the integers and counts the frequency of each will suffice. Once you have the frequency table, you can do further analysis if required. (From a frequency table, you can calculate the mean, median, standard deviation, and mode of the data, and see whether it ... 1 Although, I do think it might be more fitting to ask somewhere else, I will try to answer this from a mathematical perspective. As you noted, counting subsequences can get impractical, and doesn't, in a reasonable pratical environment, give ideal results, instead a varity of other techniques can be used. These are known as randomness tests. Your example is ... 3 In a random sequence of$n$tosses of a fair coin, the expected total number of runs is$\frac{n+1}{2}$the expected number of runs of exactly length$L$is$\frac{n+3-L}{2^{L+1}}$when$L \lt n$the expected number of runs of exactly length$n$is$\frac{1}{2^{n-1}}$An approximation to this is to say that: the total number of runs will be about ... 0 Assume that our prior information is that the program randomly generates Y with prob$p$or N with prob$1-p$, where our distribution for$p$is uniform on the interval$[0,1]$(meaning we have no idea what the parameter is). Suppose we then carry out trials and the computer returns Y$m$times and N$n$times. The probability of this result with parameter$...

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(I've revised my answer to account for the authors' use of the Dirac delta function, which I had misinterpreted as a Kronecker delta function. This is sometimes called a delta-correlated random process.) The authors appear to apply the following simulation procedure (for heuristics, see here): If $\{X(z)\}_{z\in\mathbb{R}}$ is a continuous Gaussian ...

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This is a classic problem in probability. The distribution for $N$ the first "hit" is called the geometric distribution. See the following related question: Calculate expectation of a geometric random variable

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Consider the sequence $a_{n+1}:=a_n+0.001$, $n\in \mathbb N$, where $a_1=0.001$. Obviously, this sequence does not generate random values.

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Unless there is some definition that tells us what numbers are random and what numbers are not, your question is meaningless. If you mean "Have these numbers been chosen randomly?", the answer is "probably not".

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Let suppose you have chosen a particular spins using a random number. You have calculated the energy of this microstate beforehand and let it be denoted by E1. If the spin is flipped then it will affect its neighboring spins and let suppose the energy of the new microstate is E2. If E2 And for the case when E2>E1 ,as the system has the same probability to ...

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Sketch. Let $S$ be the distance between the points, and let $X$ be the distance of the first point from the center of the circle. Then compute: $F_{S|X}(s|x)=P(S \ge s | x)$ $F_S(s)=P(S \ge s ) = \int F_{S|X}(s|x) \, f_X(x) \, dx$ $E(S) = \int_0^\infty (1-F_S(s)) ds$

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I suspect you are wondering if you could begin arbitrarily at a point somewhere within the infinite pool of decimal numbers and then start creating One Time Pads (OTP). Although it would be hard to detect by most people, and it does emulate randomness, it's not truly "pure random" because it does technically have a discernable pattern. It would fool most, ...

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There are long lists of digits for $\pi$ online. You could write a computer program that calculates different types of statistics to test your hypothesis. One interesting statistic could be to measure $$P(X_{i+1}=1|X_{i},\cdots,X_{i-n})$$That is: the conditioned probability of one bit being $1$ given that we know the binary number of length $n$ preceding ...

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$\pi$ has an infinite number of digits. Thus a book listing all the digits of $\pi$ would be infinitely big. So the room containing both you and the book would be infinitely big as well. You are in an infinitely big room, hence you are free.

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What does "random" mean? In one natural sense (or class of senses, rather) coming from computability theory, your sequence is not random: even though it's hard to notice it, there is a pattern to the bits. While it's clear that in some intuitive sense "$\pi$ mod 2" is somewhat random, and "$\pi$" is somewhat less random, I suspect it will be very difficult ...

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You should be precise about what you mean when you say "random", since "no discernible pattern" is still ambiguous. You should read about pseudorandom number generators, because that may be the kind of randomness you want. As for $\pi$, it isn't even known whether $\pi$ is a normal number in any base.

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