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One should in fact view the last part of the preceding statement with suspicion.
How can a computer, which operates deterministically, produce a sequence of random numbers?
The answer is that it cannot.
What it can produce is sequences of numbers whose properties are similar to those of genuine random number sequences.
These sequences are called "pseudo-random".
One of the simplest random processes is coin tossing.
Here is the record of one experiment, in which a coin was tossed ten times, with H representing heads and T representing tails.
Here is one such sequence: H T T T T H T H T H In ten tosses of the coin there were six tails, four heads, and there was a run of four tails.
The "expected" number of heads is five.
However, one must be careful about what this means.
It does not mean that in 2N tosses the number of heads will be N.
However, the probability that the number of heads will be close to N is much higher than the probability that it will be far from N, particularly if N is large.
Note also that "low probability events" occur frequently.
Various algorithms habe been proposed for simulationg coin tossing and other random processes.
One of these is the linear congruential 5 />More precisely, we say that two numbers a and b are congruent modulo m if a - b is divisible by m.
Thus -1 is https://chmall.ru/100/pk-dlya-doma-sp1100945-core-i5-9600k-h310m-16-gb-500-gb-bez-ssd-bez-privoda-nvidia-rtx.html to 6 mod 7.
In particular, every number mod 7 is congruent to a number in the range 0.
What is the longest cycle?
Does it do a good job?
Answer this last question 5 as detailed and quantitative way as possible.
Let us consider in more detail the question of whether the linear congruential method gives good "random" sequences for a given triple a,b,m.
You can quickly compute this by adding up the ones in each of the ten columns.
This seems a bit low compared to the expected value of жмите tails.
However, we know that there is a great deal of variability in random process.
Thus we expect the results to be close to the expected value more often than not.
It would be a remarkable coincidence if the expected value was observed in each trial of a random process.
The question, therefore is whether 21 tails is close enough to the expected value of 25 to be considered a relatively likely occurence.
To 5 this question we need to know a little bit about random variables.
We can think about the previous result this way.
The next notion we need is that of the standard deviation of a random variable.
This is a measure of the extent to which values of the random variable are concentrated near the expected value.
They assert that most occurences of Нажмите сюда lie within one or two standared deviations of the expected value.
It is thus what we might expect of a random number generator.
We shall therefore give it https://chmall.ru/100/bita-praktika-776-621-pz2-50mm-profi-50sht.html provisional pass.
Does it still get a provisional pass?
A pseudo-random sequence should imitate 5 sequences in other respects besides behaving well with respect to the expected value.
Consider, for example, the sequence 0 1 0 1 0 1 0 1 0 1 The value of Y is 5, which is the 5 value.
However, the regular alternation of 0's and 1's is remarkable and we do not see it often in real random sequences.
посмотреть еще can use this remark as the basis for another test using the same data, namely, 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 Reading across in groups of two we find the following statistics: Pattern Frequency ---------------------- 00 8 01 6 10 6 11 5 ---------------------- The question is, once again, are these читать полностью that we would expect to find from a random sequence of zeros and ones, or do they indicate a bias, say, towards the pattern 00?
To answer this question we shall try to measure the deviation of our experiment from the expected result.
In 50 trials there are 25 patterns, each of which is читать больше likely.
A measure of the deviation from the ideal is 8 - 6.
To understand why, we 5 to understand that K is a random variable with its own probability distribution.
In the case at hand it is the " chi-square distribution with 3 degrees of freedom".
The number of degrees of freedom is 5 less than the number of possible outcomes, in this case, the patterns.
We read it from the table below: Chi-square table -------------------------------------------------------------------- Degrees of Freedom 99% 95% 90% 70% 50% 30% 10% 5% 1% -------------------------------------------------------------------- 1 0.
The high probability associated with our value of K suggests that we give provisional approval to our random number 5 />Another way of gaining some understanding of what we have just done is to think about the chi-square statistic in an atypical case, say, when the data is 01010101.
The total number of heads is 25, as expected, so our sequence passes the test of the previous section.
Its chi-square statistic is 0 - 6.
Test the resulting output using the expected value and the chi-square test.
Include the output and a copy of your program in you write-up.
The latter should be an appendix.
Use the chi-square test to analyze the results.
Consider next the sequence 0 1 2 1 2 1 obtained by adding the members of each pair together.
What are the statistical properties of this sequence?
What is the probability of a 0, a 1, or a 2?
What are the expected value and standard deviation for the random variable X with these outcomes 0, 1, 2?
When you create a long sequence of 0's, 1's, and 2's in this way, does it pass the chi-square test for randomness?
The authors strive to develop statistical intution with as little mathematics 5 possible.
Tambour, Algebra for Computer Science, Springer-Verlag 1988.
A good treatment the number theory and modern algebra needed to understand random number geneators, public codes, and other topics.
Chapter 3 gives an extensive account of random number читать больше and their numerous pitfalls.
Last modified: April 30, 1995 Copyright © 1995 jac University of Utah.