Questions relating to (pseudo)randomness, random oracles, and stochastic processes.

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4answers
2k views

Random variables: How would you explain it to a beginner?

Different types of random variables: (discrete) Binomial, hypergeometric, geometric, Poisson (continuous) Uniform, normal, exponential Random variables are very useful tools when solving simple and ...
3
votes
0answers
182 views

Simulation and Stochastic Processes

Supposing we want to take a sample from the distribution $p(x)=cp^*(x)$ where $c$ is the normalization constant and $p^*(x)$ is given by $$p^*(x)=0.5\exp(-(x-\mu_1)^2)+0.5\exp(-(x-\mu_2)^2).$$ ...
4
votes
1answer
415 views

Conditional Expectation and the floor function

I have a piece of code the produces random integers. ...
3
votes
2answers
104 views

Existence of a normal computable infinite pseudorandom sequence

Is there any computable infinite pseudorandom sequence of 0's and 1's which have been proven to be normal?
0
votes
1answer
159 views

Algorithms and Simulation

Supposing we want to take a sample from a $N(0,1)$ distribution and i can take a sample from a $N(0,σ^2)$. (a) Construct a disposal/rejection algorithm with function $N(0,σ^2)$, which generates a ...
1
vote
3answers
1k views

Generate a set of random numbers with an average evenly distributed between two given values

1) I generate 1000 random numbers between 0 and 10 and take the average. If I do the above action "many" times the resulting average values will be a normal distribution over 0 to 10. Correct? What ...
0
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1answer
111 views

Test for randomness

I'm trying to write a program to compute a metric for the entropy in files to determine a probability that the file is compressed or encrypted. Compressed and encrypted files have very, very, very ...
0
votes
1answer
321 views

Choosing a random natural number with bijection with rationals

It's said that you can't choose a random natural number. But what if you take a bijection between the natural numbers and, say, the rational numbers in the unit interval, and then choose a random ...
2
votes
0answers
101 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
2
votes
1answer
3k views

Expected value in collecting a set of coupons

There are $k$ types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type $i$ with probability $p_i$ s.t. $\sum\limits_ {i=1}^kp_i =1$. If $n$ ...
8
votes
3answers
685 views

How do we check Randomness? [duplicate]

Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can ...
3
votes
3answers
247 views

Reducing an equation related to logarithms, $\pi$, and probability

In this question, I mentioned that, assuming the digits of pi are independently-random, then at some point in pi's expansion there will be a sequence of one million consecutive 0's. I decided to ...
5
votes
2answers
352 views

Does an infinite random sequence contain all finite sequences?

If we have a finite alphabet, with each letter having a non-zero probability of being selected, will an infinite sequence of letters selected from that alphabet contain all finite sequences of letters ...
1
vote
1answer
97 views

Help understanding conditional probability

Hi I'm having a hard time wrapping my head around this particular problem. Suppose the lifetime of a shirt bought from Sears, in days, is a non-negative random variable $L$ with probability mass ...
2
votes
1answer
55 views

Bound on the probability that noise changes the majority value of random bits

I have the following problem. I have a vector of size $N$ in $\mathbb{F}_2$ containing exactly $m$ zeros and $n$ ones with $m>n$. Then, a random noise is applied on each bit independently such ...
5
votes
1answer
727 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
4
votes
1answer
514 views

Generate a set of random numbers with a normal distribution

I am trying to generate a set of N random numbers where the set has a normal distribution. I'm currently using a brute force approach: Randomly select N numbers from a normal distribution. Check ...
1
vote
0answers
113 views

Maximum Likelihood Estimator of SNR for a Known Signal Superimposed in AWGN

I would like to evaluate the Maximum Likelihood Estimator for the SNR of a given signal: $ x(t) = as(t-\tau) + n(t) $ Under the following assumptions (This is the model of Radar Signal): The input ...
0
votes
1answer
517 views

doubt on iid distribution vs uniform distribution

I am a bit confused when I read "iid distribution". It looks to me like what is called "uniform distribution" i.e. a distribution of probability that is constant in a range. Am I correct in thinking ...
1
vote
3answers
904 views

Create a new pseudo-random number based on a seed, using simple formulae

I'm trying to create a function that takes two integer inputs (one < 30, one < 15), and which creates a pseudo-random value between 1 and 50. My first attempt is something like this: sum the ...
2
votes
3answers
396 views

Can a random number generator with a given number of bits have an arbitrarily long period?

Is there any upper limit for the period of any pseudo-random number generating algorithm with a fixed number of bits? I know there are some algorithms with a very large period, such as MT19937, but I ...
2
votes
2answers
105 views

One person's long number sequence: is it always pseudo random?

Nice to see you! I heard that computers cannot make real random sequences but only pseudorandom sequences. Is making random sequence that hard? Why? If pattern was found, changing a digit not ...
1
vote
2answers
119 views

Questions about averaging

i have some trouble with averages. Here are two questions rolled in one: why is : $$\frac{\prod _{n=1}^N \left(1-\text{rnd}_n\right)}{N} \neq \prod _{n=1}^N \frac{1-\text{rnd}_n}{N} \mbox{where ...
0
votes
1answer
76 views

What is name of “random boolean” algebra with set containing 0, random, and 1?

I imagine an algebra on the set of three values with an addition operation like this: ...
1
vote
1answer
125 views

Pearson's Correlation, for comparing a PRNG?

Other than uniformity tests on random numbers of which can be done with other methods, I had wondered if the result of the Pearson Product Moment Correlation function would be an effective means to ...
2
votes
2answers
123 views

Simplifying a Double Random procedure

Thought this would be a nice puzzle for some. :) First, I take a random number from 1 to 10, then take that number and multiply it by 5, we'll call this result 'threshold'. Then I take another new ...
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votes
1answer
1k views

how random is ( rand() > rand() ? true : false )

I was going to post this question on SO but I suspect it needs mathematical treatment. I need to make a decision(True or False) while running a simulation and I decided that this particular decision ...
1
vote
1answer
429 views

Signal extraction from multivariate normal

Define: $y= \theta + \varepsilon + a,$ where $a$ is a choice variable in a behavioral economic model, with equilibrium solution $a^e$, and $\theta$ and $\varepsilon$ are independently distributed ...
3
votes
0answers
746 views

Notation for sampling random variate

Is there standard notation for sampling a value from a probability distribution? Like, if I had a random variable $X$, setting $x$ to whatever value I happened to sample from $X$ on this occasion? I ...
20
votes
7answers
6k views

Generate a random direction within a cone

I have a normalized $3D$ vector giving a direction and an angle that forms a cone around it, something like this: I'd like to generate a random, uniformly distributed normalized vector for a ...
1
vote
2answers
172 views

Expected attempts before collision in multiple random numbers spaces

I am generating random 64-bit numbers, in groups of 1000 numbers. How many groups can I expect to generate before there is a collision within one of the groups? Again, I don't care if anything in the ...
0
votes
1answer
210 views

Is the average of many “random” numbers useful information?

Ok, so I found this site: http://tweetcracker.com/. Essentially, people just tweet 10 digit numbers in hopes it is the correct number (like lottery, except free). I heard that if you took all the ...
0
votes
2answers
140 views

Drawing random values from a distribution

If I have a set of $n$ elements, and I want to assign to each-one some value $\phi$, drawn at random from a distribution $f(\phi)$ such that $\int_0^1f(\phi)\;d\phi\:=\:1$ Does this mean that the ...
0
votes
2answers
2k views

Calculate the probability of a simple event

I'm beginning to study probability and an exercise in the study guide that asks me to calculate: What is the probability that the month January, of one year randomly selected have only four Sundays? ...
2
votes
1answer
125 views

Probabilistic ordering

I want to characterize the probabilistic ordering of some (random) variables without going into a parametric from of the variables themselves. I couldn't easily find any theory for this and I am not ...
0
votes
1answer
144 views

How to measure the amount of uncertainty

What are the possible measures of uncertainty for a discrete variable X=(x1, x2, ... xn), where values are defined by the alphabet - xi ∈ A, given probabilities p(xi) = P(X = xi) change over time? ...
2
votes
2answers
632 views

Types of divergence

My teacher said there are two main ways a sequence can diverge, it can increase in magnitude without bound, or it can fail to resolve to any one limit. But maybe that second kind of divergence is too ...
5
votes
2answers
1k views

Random number generation inside an interval based on cdf (Zipf and Exponential)

Consider for example the Exponential distribution with c.d.f. $F(x) = 1-e^{-\lambda x}$. $F^{-1}(x)$ would be inverse cdf (quantile function). If I generate y=F−1(x) with x uniformily distributed on ...
1
vote
1answer
316 views

How do I go about calculating the entropy level of this algorithm?

I have a set of items. These items are (pseudo)randomly placed into buckets. The buckets are ordered and items placed in them are ordered. After all of the items are placed in buckets, the items ...
0
votes
1answer
106 views

why the increment doesnt affect the randomness?

I'm doing some homework and I need to answer why the increment (b) doesn't affect randomness in the mixed congruential method. The formula is $$X_{n+1} \equiv (a X_n + b) \mod m$$
4
votes
3answers
345 views

Is it practical to use infinite continued fraction to generate random numbers?

I observed the pattern of this irrational number: $$\sqrt{1 + \sqrt{2}}$$ and realized that each element $a_i$ occurred very randomly. For the first 100 elements, this is the result: ...
0
votes
1answer
221 views

Generating a random Eisenstein integer matrix whose inverse has Eisenstein integer entries

Thanks to a question I previously asked, I realized that a Gaussian integer matrix should have a determinant of $\pm 1$ or $\pm i$ for it to have an Gaussian integer inverse. From that, I gather that ...
1
vote
2answers
132 views

terrain generation help

I'm trying to make a 3D terrain generator. In doing so, I decided that I would use basic rectangles and then just turn them by having 4 points, 1 on each side, then turn the rectangle to fit in those ...
2
votes
5answers
3k views

Generating a non uniform discrete random variable

I have an array with $N$ elements. I want to sample elements pseudo-randomly in the array in a controllable fashion. For example, I would like to sample the elements such that element $x$ is sampled ...
4
votes
3answers
556 views

What's the chance of an explicit series of integers in a limited random distribution?

Say I collect 40 perfectly random integers between 1 and 400. What's the chance that any integer is repeated consecutively six times in such a random draw? What I'm looking for is the chance of ...
7
votes
2answers
548 views

For any irrational number such as pi, would any sequence of length n appear in its decimal places?

If pi is an irrational number that goes on infinitely forever, does it mean that I can get any sequence of numbers of any length, and somewhere in the decimals of Pi, this sequence will exist. Eg. ...
1
vote
1answer
125 views

Lower bounds on the probability that one random variable is greater than a set of others

Let $X_1, \cdots, X_n$ be $n$ random variables (not necessarily independent) such that $E[X_i] > E[X_j]$ whenever $i < j$. I am interested in obtaining lower bounds on the following probability: ...
3
votes
3answers
248 views

Definition of random

Suppose that you has to guess given a set of numbers If they are random. The mathematical expectation Is there a definition of randomness that allow this prove/test? Is even possible? if so: How ...
2
votes
1answer
90 views

Meaning of randomness in space

I am a non-math person and have a question about randomness: When generating elements from a finite set using some algorithm, it is clear what it means when saying that elements should be randomly ...
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2answers
1k views

Non repeating random number generation with x(i+1) = x(i) + increment mod m

I have to generate millions of non-repeating random numbers and came across this equation: $x_{i+1} = x_i+c \space(mod \ m)$, where c and m are relative primes and $m \geq total\ to\ be\ generated$. ...