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

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1answer
211 views

Period of least significant bits of linear congruential random number generator

On a practice exam for a course on stochastic simulations I encountered the following question: Show that the least significant $n$ bits must repeat with a period $2^n$ for a congruential random ...
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3answers
319 views

Get random $x, y, z$ point inside a sphere?

What is the easiest way to get a random Point3D inside the sphere where is red?
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1answer
629 views

What does Linear Congruential mean?

How does one interpret the terms "Linear" and "Congruential" as in a "Linear congruential RNG"? I am used to linearity by $f(ax)=af(x)$. This does not seem to me to hold true in this case ($\bmod$). ...
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4answers
13k views

Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...
2
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2answers
67 views

Expectation of random callers distribution

In a country with $N$ people, if everyone calls one random person in the country, what is the expected number of people who dont receive a call? If $f(m,n,a,b)=$Expected fraction of people who ...
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0answers
133 views

Correlated Random Numbers Between A Range

I am trying to generate random numbers within a range say 57 to 107 which are correlated by a factor of 0.8. The numbers are for illustration simplicity only.
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2answers
407 views

Are probabilities proportional to the distance traveled in a random walk? What if the initial position is a bit biased?

A marker is placed at zero on the number line and a fair coin is flipped. On each flip we move one unit to the right. If it lands on heads, the marker is moved one unit up. If it lands on tails, the ...
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1answer
880 views

runs above and below the mean test

In the runs above and below the mean test of independence, the mean is MEAN = (2*n1*n2)/N - (1/2) So here, are n1 and n2 runs above and below or ...
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1answer
199 views

If W is a random matrix with variance $\mathbb{E}[W W^{T}]$, what's $\mathbb{E}[W^{T} P W]$?

I know quite a few identities about quadratic forms of random vectors, but I'm having difficulty coaxing something out of this quadratic form of random matrices. Suppose I know $\mathbb{E}[W W^{T}]$ ...
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1answer
124 views

random variable transformation

I'm having trouble with the following random variable transformation: $Y = X^2 + X$ I am looking for the pdf of Y. I tried the following method: $p_Y(y) = \int_{X} p_{Y|X=x}(y)\cdot p_{X}(x)dx$ and ...
6
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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 ...
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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 $$p^*(x)=0.5\exp(-(x-\mu_1)^2)+0.5\exp(-(x-\mu_2)^2).$$ ...
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1answer
421 views

Conditional Expectation and the floor function

I have a piece of code the produces random integers. ...
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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?
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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 ...
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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 ...
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1answer
112 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 ...
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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 ...
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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 ...
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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 $n$ ...
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3answers
694 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
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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 ...
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2answers
356 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 ...
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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 ...
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1answer
56 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 ...
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1answer
754 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?
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1answer
515 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 ...
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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 ...
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1answer
534 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 ...
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3answers
926 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 ...
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3answers
397 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
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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 ...
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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 ...
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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: ...
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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 ...
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2answers
124 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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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 ...
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1answer
436 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
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0answers
757 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 ...
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7answers
7k 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 ...
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2answers
173 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 ...
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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 ...
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2answers
142 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 ...
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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? ...
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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 ...
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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
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2answers
641 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 ...
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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 ...
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1answer
317 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 ...
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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 $$X_{n+1} \equiv (a X_n + b) \mod m$$