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Questions tagged [random]

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

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When a bivariate probability distribution can be the probability distribution of $(X_1, X_1-X_2), (X_2, X_2-X_1), (-X_1, -X_2)$

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following questions: (1) Are there necessary conditions on the cumulative distribution function (CDF) ...
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Generalized Polya's Urn

In Polya's urn, we have $b$ black balls and $w$ white balls at time $t$. At time $t+1$, we have $b+1$ black balls with probability $\frac{b}{w+b}$ and $w+1$ white balls with probability $\frac{w}{w+b}$...
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Repeating random choice

Let's say I have 10 shirts (A to J) and I'd like to randomly wear a different one each day. So, the one I wore yesterday (let's say shirt J) has 0% chance of being chosen today; The one I wore the ...
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Calculating expectations of concentrated random variables of bounded-differences type

Is there a nice general way of calculating the expectation variable for which I can derive concentration bounds using the method of bounded differences? I have seen quite a few application of the ...
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1answer
23 views

Generating Two Independent Standard Normal Variables

Image of problem I have a problem with this explanation, the book says we can generate Y1 Y2 from X1 X2 but when we want to answer questions about the distribution of X1 X2 we take arctan which ...
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1answer
34 views

st.petersburg paradox in python

I have been trying to do the St.Petersburg paradox( the player wins 2 dollars if tail appears on the first toss, 4 dollars if heads appear on the first toss and tails on the second, 8 dollars if heads ...
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Using randomly selected points uniformly distributed on the interval (0,1) find the volume of the unit sphere

I've been given this extra credit assignment to do in Matlab, but I don't understand the question. The part that I'm not clear about is where I have to satisfy the equation x+7+z>1. Can someone ...
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Thinning operation on a spatial binomial point process (BPP)

It is known that the thinning operation performed on a Poisson point process (PPP) results in a non-homogeneous PPP. Now, if we have a spatial BPP with N points distributed uniformly over some region ...
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Generating a random sparse hermitian matrix in Python

I'd like to find a way to generate random sparse hermitian matrices in Python, but don't really know how to do so efficiently. How would I go about doing this? Obviously, there are slow, ugly ways to ...
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1answer
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Expected value in multiple rounds

I am doing an exercise that sounds like: The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land ...
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Help calculating second order distribution for a continuous stochastic process

I'm really stuck on this problem, and appreciate any help. Problem: Let $\gamma$ be a random number, chosen with uniform probability in the interval $[0,2\pi]$. We define the stochastic process $X(t)...
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Chi-Squared (observed) formula

Chi-square distribution shows how far the observed values are from expected values. [NIST SP 800-22] (https://nvlpubs.nist.gov/nistpubs/legacy/sp/nistspecialpublication800-22r1a.pdf) employs Chi-...
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Create unique one to one number from 2 numbers

A smilar question has been asked before Create unique number from 2 numbers. is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 ...
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Generate random numbers with conditions using maxima (cas)

I want to generate (multiple) random numbers from a given number class with extra conditions. (for automatic generation of exercises) e.g.: generate "a" out of natural Numbers which are dividible by ...
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Working out expectation of a random sample.

I have the problem: Let $X_1, X_2, X_3, X_4$ be a random sample from a population that has mean $μ$ and variance $σ^2$. Find $\mathbb E[(X_1-X_2)^2]$ and hence the value of $k$ such that $T ...
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Two aspects of randomness

Consider a random sequence of integers 1, 4, 3, 8, 2, 5, 3, 8 ... The only sufficient condition for the sequence to be random is its unpredictability ie. probability of any number coming next ...
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2answers
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Probability of picking 3 playing cards that belong to different suits

If you have 52 playing cards with 4 different suits (13 spades, 13 clubs, 13 hearts, and 13 diamonds), what is the probability of picking 3 random cards that are all from a different suit? I tried ...
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What is the mean of a random number with one exclusive bound?

If I have a random number between 0 and 1 inclusive, the mean is 0.5. If both are exclusive, the distance from 0.5 to 0 and 0.5 to 1 are the same, so 0.5 is still the mean. But if 0 is inclusive and 1 ...
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Expected value and variance of random variable on some terms(dates?)

I have a exercise that sounds like this: Find the expected value and the variance of the random variable X with natural values in terms of: a)generating function P b)generating function Q Excercise ...
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Expected Length of Arc in Randomly Divided Circle.

Let's say theres a circle of unit circumference m, and the circle is divided by k points all placed randomly. The points will divide the circle up into arcs. What is the expected length of any arc in ...
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1answer
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Why does the Monte-Carlo Method Work?

I've been reading about the Monte-Carlo Method and how it is much simpler for computers to use the Monte-Carlo Method to guesstimate solutions to complex problems like the Standard Model. It is ...
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Probability - Similarity Rate in the random Signal

I have a random signal (vector) that consists of random variables varying between -2 and 2. I want to know how many similar patterns do I have in this random signal. In order to achieve this, I select ...
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Can a sum of lagged variables random process be a Markov chain

Consider the following random process: $X_0=0$, $X_{t+1}=X_t+Z_tZ_{t-1}...Z_0$ for $t\geq0$, where $Z_t$ are i.i.d. random variables with normal distribution. In this process: $Z_0=X_1$, therefore ...
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Generating vector inside a $n$-sphere

I want to generate k n-dimensional vectors which are all inside a r-radius n-sphere and the most important : I want something uniformly distributed inside the n-sphere. My initial idea is to generate ...
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1answer
51 views

What would be the chances of solving a standart 3000 piece puzzle completely at random?

I LOVE puzzeling, but this one question always bothered me. And for that I am not good at maths, I couldn't really work it out. Here is the problem: Say I have a 3000 pieces puzzle and I blindfold ...
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1answer
46 views

random walk with finite range

Let $X=(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. $\mathbb{Z}$-valued random variables satisfying the following conditions: a) For all $n \in \mathbb{N}$ and $k \in \mathbb{Z}$, we have $\...
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1answer
23 views

Counting tree nodes that end probabilistically

Suppose we have $N$ root nodes that can branch into $N$ new nodes every level. There is a maximum of 64 levels. However, on any given step, each node has a chance $P$ of terminating. What is the ...
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Soft Question: Concept of Density of a Poisson Point Process (PPP)

Consider the following code: lamda= 0.00001 AreaR=3*10^3; M= poissrnd(lamda*AreaR^2); Is the generated PPP of density lamda...
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1answer
43 views

Probability and Computer science question

We are given a function rand() that returns a random number from the segment [0,1], how can we use this function to create a size $100$ uniform array, of exactly $50$ $0's$ and $50$ $1's$.
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1answer
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When is an i.i.d Bernoulli Process indepent increment?

I was reading anarticle that i.i.d bernoulli process are markov and independent increment process are markov too. So, I was wondering if an i.i.d. bernoulli process can be independent increment.
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Is the product of two random variables finite if the expectation of the random variables squared is finite?

If $E X_t^2 < \infty$ and $E Y_t^2 < \infty$, then is $E X_t Y_t$ finite? I am thinking yes, because of Cauchy-Schwartz, since $E X_t Y_t - E X_t E Y_t$ is an inner product, so it's less than ...
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True random generation given radioactive entropy

I have been reading various sources for experimentation with truly random numbers. As I understand, it is impossible for a computer to generate a "truly random" number as they are deterministic in ...
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1answer
28 views

How to form an equation for probability based on the following graph?

The y-axis is the probability density function of x( a continuous random variable). Here's the solution I tried: Is this correct?
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1answer
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Generating 5 random numbers less than 100 that add up to a number, 180 for example

I have found an answer (this) showing how to generate random numbers that add up to a number, but I need those numbers to be less that 100. How might I go about this?
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Identifying bias from constraining CSPRNG output

Let's say I have a sequence of random numbers between $0$ and $1$ given by $[x_1, x_2, \dots, x_n]$ from a CSPRNG. The distribution is uniform. Now I permute my random numbers such that it follows a ...
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1answer
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Balls and Bins problem max size of bin

I need to make a program which satisfies the below (my question is not about code): The Balls’n Bins-problem deals with the experiment of randomly distributing a number of balls into an equal ...
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Randomized Submatrix of a Sparse Matrix

I have a sparse square matrix $A$ with size $n \times n$ and number of nonzero entries $nnz$. The goal is making a sub-matrix $B$ with $s$ nonzeros which are randomly chosen from $A$. Duplicates are ...
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1answer
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Let X be distribution over N (the set of non-negative numbers), with mass P(X=i) = a/2^i, what is the value of a?

I am struggling with solution for following problem part of course about probabilities random variables, seek your kind help to show how to solve it, Let $X$ be distribution over $N$ (the set of non-...
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Sample from random normal with sliding mean

I have a uniform random variable $x$ and a normal random variable $z = \mathcal{N}(x, \sigma)$ (i.e. the mean is given by $x$). How can I draw samples $(X, Z)$ such that they correspond to their ...
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The independence of two random uniform distribution random variables

$y_1 = x_1 + x_0$; $y_2 = x_2 + x_0$. Suppose that $x_1$, $x_2$, and $x_0$ are independent with each other. They all follow the uniform distribution in $[0, 1]$. Then, I want to know if $y_1$ and $...
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Random walks in different directions?

Assume a person random walker takes equal steps to the right or left with equal probability. Probability that taking n steps, the person walking will be displaced 1 standard deviation or greater in ...
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How many roots exist for $y=sec(x)$

In the interval $( - \pi ,\ \pi ]$.There are 2 roots exist mentioned in the book. Could anyone please explain how? Exact question from book : Let $Y=\sec X$ .Compute $f_Y(y)$ in terms of $f_X(x)$ ....
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Simulate a discrete random variable

We have a discrete random variable $X$ with the following probability distribution \begin{equation*} p(X=i)=p_i,\quad i=1,2,\ldots 1000, \quad \sum_{i=1}^{1000}p_i=1. \end{equation*} How we can apply ...
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Probability of having a link in union of Erdos Renyi random graph

We have two Erdos-Renyi random graphs, $G_1$ and $G_2$, generated with probability $p_1$ and $p_2$, respectively. If we take the union $G_1$ $\bigcup$ $G_2$, we obtain another Erdos-Renyi graph, $G_3$...
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probability after ten draws

I would like to know if my partial solution is appropriate: We have ten urns, in each urn we have 2 balls: in 1st urn - 2 balls of number 1 in 2nd urn - 2 balls of number 2 . . in 10th urn - 2 ...
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Question on Random Variables from the textbook “Mathematical Statistics and Data Analysis” by John Rice

If $U$ is a uniform random variable on $[0,1]$, what is the distribution of the random variable $X=[nU]$, where $[t]$ denotes the greatest integer less than or equal to $t$? I've tried searching ...
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Connections between the randomness of the normal distribution and Textrank?

In a TED speech on 8:40 the mathematician said that: This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages, in the same way as we used ...
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38 views

Probability two sequences of coin flips reach consecutive heads at the same time.

Two people start flipping coins. The probability of heads is 0.5 (bonus if you can do it for $p_1$ and $p_2$). What is the probability that both will hit two consecutive heads simultaneously (as ...
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Sum of Uniform(5,10) random variables to get more than 30

Let $X_i$ be i.i.d. $Uniform(5,10)$, and let $Y_t = \sum_{i=1}^t X_i$. Let $T = \mbox{inf}\{t:Y_t \geq 30\}$, what is $\mathbb{E}[T]$? At first I thought this was similar to Choose a random number ...
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Reachability of points on Manifolds

I was thinking with a friend that on a surface some points are more reachable than other.In the sense that their average distance to the other points is lower. e.g. suppose that we have a circle in a ...