Properties of sequence of random numbers and how to generate random numbers?

In summary, generating random distributions involves using algorithms and methods to create pseudorandom numbers that mimic genuine randomness. The most popular method for generating uniform random numbers is the Linear Congruential Generators, but it is important to choose proper values to ensure a truly random and uniform distribution. If the conditions for non-zero c are not met, it can lead to a less random and uneven distribution.
  • #1
WMDhamnekar
MHB
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Hi,
Would any member of Math Help Board explain me the highlighted area in the following paragraphs?
1645111239487.png

1645111418925.png

Generating Random Distributions
Now the only missing thing in previous cases is how would one generate a Uniform random, Normal random distributions. We therefore look to cover algorithms to generate such Uniform random distributions and also methods to transform these to other distributions such as Normal Distributions.

The numbers that we will be generating in R for these simulations are actually called pseudorandom numbers which are not truly random but pretty good at mimicking genuine randomness. We generate them using deterministic algorithms such that they approximate the properties of sequence of random numbers.

Uniform Distribution

Uniform Distribution is a continuous distribution abbreviated as U(a,b). Where the probability density function is

1645111949987.png
The most popular method for generating such numbers is the Linear Congruential Generators. The LCG is generated using-

1645112033250.png


Thus if we choose a x₀ also called the seed. We can generate m-1 unique Uniform Random Numbers between [0,1] . Though a good estimator must follow certain conditions for non zero c.
1645112121455.png


If the above conditions for non-zero c are not fulfilled by a good estimator, what are its effects and results?
 
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  • #2
If the conditions for non-zero c are not fulfilled by a good estimator, it can cause the LCG to generate a small number of distinct values and then repeat those values in a pattern. This would result in a much less random distribution than desired. Additionally, if the values of c, a, and m are not properly chosen, the LCG could produce an output that is not uniform. This means that the probability of some values might be much higher than others, creating an uneven distribution.
 

Related to Properties of sequence of random numbers and how to generate random numbers?

1. What are the different types of properties of a sequence of random numbers?

The main properties of a sequence of random numbers are randomness, uniformity, independence, and reproducibility. Randomness refers to the unpredictability of the numbers in the sequence, while uniformity means that each number has an equal chance of being selected. Independence means that the numbers in the sequence are not influenced by previous numbers, and reproducibility means that the same sequence can be generated multiple times.

2. How are random numbers generated?

Random numbers can be generated using various methods, such as physical methods like rolling dice or flipping coins, or computational methods like using algorithms. These algorithms use a seed value, which is a starting point for generating the sequence of numbers. The seed value can be a specific number or a combination of factors such as time or user input.

3. What is the importance of having a good random number generator?

A good random number generator is essential in various fields, including statistics, cryptography, and simulation. In these fields, it is crucial to have truly random and unbiased numbers to ensure accurate and reliable results. A poor random number generator can lead to biased data and incorrect conclusions.

4. Can random numbers be predicted?

No, truly random numbers cannot be predicted. While some patterns may seem to emerge in a sequence of random numbers, it is impossible to predict the next number without knowing the algorithm or seed value used to generate them. Additionally, a good random number generator should produce numbers that are statistically independent, making it impossible to predict future numbers.

5. How can the quality of a random number generator be evaluated?

The quality of a random number generator can be evaluated using various statistical tests, such as the chi-square test and the runs test. These tests analyze the sequence of numbers and check for any patterns or biases. Additionally, the generator can be tested for statistical properties such as uniformity, independence, and reproducibility.

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