Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln ( N ), where ln ( N ) is the natural logarithm of N. For example, among the positive integers up to and including N = 10 < sup > 3 </ sup > about one in seven numbers is prime, whereas up to and including N = 10 < sup > 10 </ sup > about one in 23 numbers is prime ( where ln ( 10 < sup > 3 </ sup >)= 6. 90775528. and ln ( 10 < sup > 10 </ sup >)= 23. 0258509 ).

The prime number theorem asserts that an integer m selected at random has roughly a chance of being prime.

Pick a random integer q, such that

If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function

The number of monochromatic r-subgraphs in our random coloring will always be an integer, so at least one coloring must have less than the expected value.

pick a random integer a in the range n − 2

The best studied example is of random walk on the d-dimensional integer lattice ( sometimes called the hypercubic lattice ).

An elementary example of a random walk is the random walk on the integer number line,, which starts at 0 and at each step moves + 1 or − 1 with equal probability.

Rather than storing random priorities on each node, the randomized binary search tree stores at each node a small integer, the number of its descendants ( counting itself as one ); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation.

The information stored per node in the randomized binary tree is simpler than in a treap ( a small integer rather than a high-precision random number ), but it makes a greater number of calls to the random number generator ( O ( log n ) calls per insertion or deletion rather than one call per insertion ) and the insertion procedure is slightly more complicated due to the need to update the numbers of descendants per node.

The random starting point should instead be selected as a noninteger between 0 and 15. 625 ( inclusive on one endpoint only ) to ensure that every house has equal chance of being selected ; the interval should now be nonintegral ( 15. 625 ); and each noninteger selected should be rounded up to the next integer.

where is an array object, the function chooses a random integer between 1 to x, inclusive, and swaps the ith and jth entries in the array.

The random Fibonacci sequence is an integer random sequence

For instance, in Kerberos, the challenge is an encrypted integer N, while the response is the encrypted integer N + 1, proving that the other end was able to decrypt the integer N. In other variations, a hash function operates on a password and a random challenge value to create a response value.

* Encode the secret as an integer s. Give to each player i ( except one ) a random integer r < sub > i </ sub >.

# Select random integer where

Here the random vector is the vector r of random returns on the individual assets, and the portfolio return p ( a random scalar ) is the inner product of the vector of random returns with a vector w of portfolio weights — the fractions of the portfolio placed in the respective assets.

Originally, the definition for relatedness ( r ) in Hamilton's rule was explicitly given as Sewall Wright's coefficient of relationship: the probability that at a random locus, the alleles there will be identical by descent ( Hamilton 1963, American Naturalist, p. 355 ).

A traditional RSA signature is computed by raising the message m to the secret exponent d modulo the public modulus N. The blind version uses a random value r, such that r is relatively prime to N ( i. e. gcd ( r, N )

Because r is a random value and the mapping is a permutation it follows that is random too.

Additionally to the scheme above, it uses another generator h of the prime group and a random number r. The commitment is set.

Consider a sphere S ( r ) with radius r. A point on the sphere is identified by its latitude φ and longitude λ, for which we introduce the random variables Φ and Λ that take values in Ω < sub > 1 </ sub > = respectively Ω < sub > 2 </ sub > =.

Paillier and Pointcheval however went on to propose an improved cryptosystem that incorporates the combined hashing of message m with random r. Similar in intent to the Cramer-Shoup cryptosystem, the hashing prevents an attacker, given only c, from being able to change m in a meaningful way.

The role of the random r ensures that two equivalent votes will encrypt to the same value only with negligible likelihood, hence ensuring voter privacy.

The class PCP refers to the set of decision problems that have probabilistically checkable proofs that can be verified in polynomial time using at most r ( n ) random bits and by reading at most q ( n ) bits of the proof.

The randomness complexity r ( n ) of the verifier is the maximum number of random bits that V uses over all x of length n.

for Brownian motion ( since the random force F < sub > rnd </ sub > is uncorrelated with the position r ).

Let an individual's increasing, concave von Neumann-Morgenstern utility function be u, let r < sub > f </ sub > be the return on the risk-free asset, and let r be the random return on the risky asset.

where t is a random variable distributed as Student's t-distribution with r. d. f.

The number of successes on each shot is a value of a random variable that has values 0 or 1, and there are 5 such random variables here.

This category covers random sources such as noisy diodes and radioactive decay.

In information theory, one bit is typically defined as the uncertainty of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

The term " Brownian motion " can also refer to the mathematical model used to describe such random movements, which is often called a particle theory.

Some viral vectors effect this in a random fashion, which can introduce other problems such as disruption of an endogenous host gene.

There are 2, 598, 960 such combinations, and the chance of drawing any one hand at random is 1 / 2, 598, 960.

There are three general categories of casino games: table games, electronic gaming machines, and random number ticket games such as Keno and simulated racing.

Some casino games have a skill element, where the player makes decisions ; such games are called " random with a tactical element ".

Belief in such a cabal is a device for reassuring oneself that certain occurrences are not random, but ordered by a human intelligence.

The codex holds considerable practical advantages over other book formats, such as compactness, sturdiness, ease of reference ( a codex is random access, as opposed to a scroll, which is sequential access ), and especially economy of materials ; unlike the scroll, both recto and verso could be used for writing.

This packing material can either be random dumped packing ( 1 – 3 " wide ) such as Raschig rings or structured sheet metal.

* In secondary structure denaturation, proteins lose all regular repeating patterns such as alpha-helices and beta-pleated sheets, and adopt a random coil configuration.

Divination can be seen as a systematic method with which to organize what appear to be disjointed, random facets of existence such that they provide insight into a problem at hand.

Leading libertarian philosophers such as Robert Kane have rejected Dennett's model, specifically that random chance is directly involved in a decision, on the basis that they believe this eliminates the agent's motives and reasons, character and values, and feelings and desires.

They assembled at certain points, such as pubs, each morning, where they were selected more or less at random by foremen.

For certain types of point sets, such as a uniform random distribution, by intelligently picking the splitting lines the expected time can be reduced to O ( n log log n ) while still maintaining worst-case performance.

There is an extensive body of mathematical theory that explores the consequences of making the allocation of units to treatments by means of some random mechanism such as tables of random numbers, or the use of randomization devices such as playing cards or dice.

Provided the sample size is adequate, the risks associated with random allocation ( such as failing to obtain a representative sample in a survey, or having a serious imbalance in a key characteristic between a treatment group and a control group ) are calculable and hence can be managed down to an acceptable level.

Many-worlds claims to reconcile the observation of non-deterministic events, such as the random radioactive decay, with the fully deterministic equations of quantum physics.

Commenting on this, other writers ( such as John von Neumann and David Bohm ) have suggested that consequently there would have to be ' hidden ' variables responsible for random measurement results, something which was not expressly claimed in the original paper.

In all such experiments order of the cards must be random so that hits are not obtained through systematic biases or prior knowledge.