Questions tagged [random]

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

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18 views

Spectrum of sum of (weighted) random matrices

Coming from statistical physics, I am interested in the spectrum of the following sum \begin{equation} \sum_{n=1}^m c_n X_n, \end{equation} where $c_n$ are non-random real numbers and $X_n$ are ...
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1answer
21 views

Complete randomness/disorder and determinism

Wikipedia page about randomness says that "complete disorder" and "true randomness" are impossible according to Ramsey Theory and Cristian S. Calude. I don't understand it. https://...
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Unbiased estimator given pdf

Let $X_1,X_2,…,X_n$ be independent samples from a distribution with pdf $f_X(x)=\frac{1}{θ^2}xe^{-\frac{x}{θ}}$ $(X≥0)$. Which of the following is an unbiased estimator for $θ$ ? I'm trying to solve ...
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2answers
43 views

Using Ito calculus to prove that $\int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^t W_s d_s$

I am busy trying to teach myself some stochastic calculus and have come across a statement that I am trying to prove. How can I prove that \begin{align} \int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^...
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23 views

Series of probabilities

Consider $Y_k=X_1+...+X_k $, where $X_k \in \mathbb{N}_0$ are i.i.d random variables and $E[X_1]<1$ $$\sum_{j=1}^{\infty} P(Y_j=j) \overset{!}{=}1 $$ How can I verify that this equation is true or ...
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1answer
58 views

Benford's law not working?

So I recently came across Benford's law and immediately tried to code it out but the answer I got was rather confusing. I think my code is correct I'm pretty sure it is but the result is just not the ...
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1answer
48 views

Simple Random Walk with equal probability of +1 and -1.

You have 1D random Walk, with +1 of probability 0.5, and -1 of probability 0.5. What is the probability that you will reach +10 but never exceed -5? Attempt: The probability of getting +10 is easy, ...
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31 views

upper bound for expected for special random variable

We throw three times independently perfect dice. For each random realisation of our throwing we have the random vector $$(x_{1}, x_2,x_3)$$ where $x_i\in\{1,2,3,4,5,6\}$ for $i=1,2,3.$ We now consider ...
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27 views

Long string of same rolls in a random sequence of dice throws

I'm not a mathematician and my knowledge of probability and statistics is limited. I came across an interesting problem, while discussing randomness in games, and honestly I don't even know where to ...
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9 views

Are the last digits of a uniformly distributed random sequence going to always appear at the same frequency approximately?

I noticed that one can sample a sequence of uniformly distributed integers (user IDs in my case) algorithmically in sizes of 10%, 20%.. etc. by only choosing those numbers whose last digits equals one ...
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22 views

Does intersection mean set theoretic point of view for independent events?

It is evident that when two events $A$ and $B$ are independent, then $P(A\cap B)=P(A)P(B)$ A good example is, "Tossing a coin and rolling a die". What is the probability that heads occurs on ...
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Do all LCG-based PRNGs suffer from predictable patterns?

I needed to produce trivial (low-quality) random integers and remembered how simple linear congruential generators were to implement from school: Went to Wikipedia, found the first example which ...
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1answer
127 views

Random Shuffle of Groups

Let's suppose that we have 54 peoples and we arrange them into 9 groups of equal size, so this means that each group will have 6 persons in it. I want to find a procedure, such that the groups are ...
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1answer
20 views

Normally Distributed and Covariance is zero [duplicate]

Let $X \sim N(0,1)$, and let $P(Y=1)=P(Y=-1)=\frac{1}{2}$. Assume $X$ and $Y$ are independent, and let $Z = XY$. Prove that $Z \sim N(0,1)$ and $Cov(X,Z)=0$ For proving $Z$ is a standard normal ...
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What is the general form of random sampling a radial direction?

In 2-dimensions, if I want to traverse to some location $\rho$ away from the origin with equal probability, I can simply sample $\theta \in [0, 2\pi]$ and apply $x = \rho \cos \theta, y = \rho \sin\...
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50 views

How do we show $P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B).$?

In some questions I have been going through here, I came across this inequality several times. $A$ and $B$ are RV. How do we show $$P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B)$$.
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Convergence of Number of Success in Bernoulli Trials

Assume the probability of success in the Bernoulli trial is $p$ with $0\leq p \leq 1$, let $S_n$ be the number of success in $n$ trials, determine whether $S_n/n$ converges. So I know how to ...
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14 views

Distribution of the Inner Product of a Gaussian Random Vector and its Reversal

Suppose I have a random variable $\mathbf{x} \sim \mathcal{N}(0, \sigma^{2} \mathbf{I})$, where $\mathbf{x} \in \mathbf{R}^{N}$, where $N$ is an even number. Let $\mathbf{x}_{r} = \mathbf{A}_{rev} \...
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18 views

Distribution of random numbers with fixed sum

I have access to a black box function $f$ that returns 4 random integers $n_1$, $n_2$, $n_3$, $n_4$ with $4 \le n_i \le 13$ and $\sum_i n_i = 25$. Experimentally, I can see that $n_1$, $n_2$, $n_3$ ...
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1answer
49 views

Normalising Low Duty-Cycle Random Data

I'm an engineer by training and developing a quantum random number generator for high-security cryptographic applications. I'm using single-photon arrival times as my source of entropy for the device. ...
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1answer
26 views

A Probability Question From HW - Why a Random Variable Has Two Distribution?

Assume we have a random number generator that can generate a random number uniformly distributed in the range $[0, 1]$. If now we want to use this generator to generate a random number X that has a ...
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23 views

Randomly guessing answers on a randomly generated test

My question stems from answering questions (on a test) whose answer order was randomly chosen (not by a human, by a computer). I’m skeptical of something I just read: Your overall score will improve ...
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1answer
32 views

strategy for game with randomly chosen numbers

Professor $X$ chooses a number $x$ between $a$ and $b$ ($a < b \in \mathbb{Z}$) and asks two other students $A_1$ and $A_2$ to pick numbers between $a$ and $b$. Whichever is closer to $x$ gets $90\...
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44 views

Sample random high-dimensional rotation, of bounded angle

I want to sample uniformly at random from all $n\times n$ orthogonal matrices $M$, subject to two requirements: $M$ is an orthogonal matrix. Loosely speaking, it represents a rotation (or rotation ...
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24 views

Random Matrix Product

Suppose I have the random matrix $G \in R^{d \times d}$. I draw each element in $G$ independently from a Gaussian with Zero mean and finite variance. What can I say about the product of $GG^T$? ...
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1answer
27 views

Expected value of time taken on random-walk on a cube

$A$ and $B$ are opposite vertices of a cube. A spider sits on $A$ and takes steps randomly along any edge connected to the vertex it is on at that moment. This situation ends when the spider makes it ...
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2answers
29 views

Given an n-sided polygon, how would you random sample points within it?

I would like to random sample points within an n-sided polygon. One idea I've thought of is to discretize the area of the n-sided polygon with triangles. I can assign each triangle a unique number in ...
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1answer
27 views

Dependence of coin tosses

We are tossing a coin $m$ times The probability of heads $\Bbb P(H)$ is anywhere in $(0,1)$. It doesn't have to be a fair coin basically. Random variables $X$ and $Y$ are defined as the total numbers ...
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34 views

Paradox of randomization and memorylessness?

I am thinking about a randomizer, namely $A$, which randoms a single-digit number $0-9$ under a uniform distribution. The machine has a limited lifespan (can generate only a finite sequence of digits) ...
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1answer
25 views

Problem about probability of observing a particular DNA sequence related to Beta binomial conjugacy

I have some questions about the following problem from Introduction to Probability by Joseph K. Blitzstein and Jessica Hwang. Problem statement A DNA sequence can be represented as a sequence of ...
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2answers
94 views

What is the probability of getting the same number on two dice?

If two identical dice are thrown simultaneously (The order of result does not matter. For example, $(2, 3)$ and $(3, 2)$ are considered same), what is the probability of getting same number on both ...
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Is there a rigorous way to differentiate between different types of randomness?

Suppose I have a deck of cards. The probability of picking the Jack of Hearts is $P(J♥︎)=\frac1{52}$. Suppose I have an ordinary, unweighted d6. The probability of rolling a $4$ is $P(4)=\frac16$. ...
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1answer
14 views

Average of the numbers given by a linear congruential generator

Show that the average of numbers given by a full cicle of a linear congruential generator with full lenght period is $\frac{1}{2}-\frac{1}{2m}$, whith $m$ as the module of the generator. I began by ...
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1answer
89 views

Statistics problem about Possibility principle

I have a question about the following problem from the first edition of Introduction to Probability by Joseph K. Blitzstein and Jessica Hwang. This is problem 68 from Chapter 4. Each of 111 people ...
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1answer
40 views

random uniform dot in triangle

$\triangle ABC$ has sides of length $45,60$ and $75$. Place a point $D$ randomly and uniformly inside the triangle. What is the expected value of the sum of perpendicular distances from point $D$ to ...
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1answer
25 views

Getting samples or subsets from a distribution

I probably missed some basic basic math class, but I'm struggling a bit with finding a solution for this marble problem. Suppose you want to divide $Y$ marbles into $Z$ categories. Every category has ...
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44 views

Algorithm to generate insecure random numbers [closed]

I would like an algorithm which can generate a list of random, uniformly distributed floating point numbers from a given seed, ideally also being able to specify the number of decimal places. The use ...
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5answers
2k views

So what really is a random variable?

I've been confused about the definition of a stochastic (random) variable for an embarrassingly long amount of time. From the Wikipedia entry on Stochastic Processes, I could gather that a random ...
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2answers
85 views

How to estimate total number of different results for a stochastic event?

Let $X$ be a random variable that has $n$ possible values $x_1,x_2,\dots,x_n$, and that $P(X=x_i)=\frac 1n ,\forall x=1\dots n$. Note that $n$ and $x_i,i=1\dots n$ are all unknown and $x_i$ is ...
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1answer
68 views

How can a random number 0-100 be generated from an initial number?

A group of people want to do a random drawing with no person involved, just grabbing an initial number from public events (ex. score of a team in a NBA game) and then make it as equally likely that it ...
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27 views

Sampling random sets of integers with bounded range

How can we randomly sample a set $S$ of $k$ integers from the range $[1, n]$ such that: The set satisfies $\max(S) - \min(S) \leq w$. Each integer in $[1, n]$ has an equal chance of being in the set....
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1answer
103 views

Is compressibility a good test for randomness of a pseudorandom sequence?

I am interested in tests and definitions of randomness of a sequence generated by a pseudo-random number generator. A similar question was asked a few years ago, and the response was to use a ...
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47 views

Pseudo Random Number Generator

I want to implement a very simple random number generator in Google Sheets to play DnD with my friends. There is a built in, but the problem is that it does not take a seed, and every time we roll it ...
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1answer
24 views

Generate 3 random numbers such that the sum is below a number, or within a range

I'm writing a program for fun which generates random colours for text. (If you're curious: https://github.com/Arunscape/acat) So, I have 3 variables $r, g$, and $b$ , each of which can be any integer $...
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1answer
34 views

Product distribution of two uniform distributions which are centered around 1

Consider the product distribution $Z = X_1\cdot X_2$ for $$ \begin{aligned} X_1 &\sim \textrm{Uniform}[1 - a, 1 + a] \quad, \quad 0 < a < 1 \\ X_2 &\sim \textrm{Uniform}[1 - b, 1 + b]...
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1answer
41 views

Family vs. Child when a girl is chosen, what is the probability that the second child is a girl, textbook clarification?

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I was able to understand the difference between selecting a child and a family, for ...
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28 views

Decoupling two-dimensional simple symmetric random walk

It is an observation that the two-dimensional simple symmetric random walk can be broken into two independent one-dimensional random walks : that is, let $S_n$ be $2$D SSRW on $\mathbb Z^2$. The ...
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4answers
212 views

What is the optimal strategy of guessing a number where closest without going over wins?

When a group of people need to decide a winner or leader between them, one approach would be that a random hidden integer is chosen with uniform distribution on $\{0, 1, ..., n\}$ and all $p$ ...
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25 views

Given a Random Process, how do I determine it is ergodic?

I am working on the following homework problem: So to determine if a process is stationary, I understand that you must prove that the mean and auto-correlation function must not be dependent on time (...
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2answers
146 views

How to calculate birthday problem for collisions of a random number generator

I'm creating a script that generates random numbers for invoices as fully described here. The current version of my script includes logic to prevent collisions, but this means that every time there is ...

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