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

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Probability of a specified sequence in a random data set

This is a problem which I have encountered while programming, but I imagine this community would be better able to solve it. Suppose we have a number, N, of boxes in a row. In each of these boxes is ...
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1answer
28 views

probility, placing balls, covariance

Can you please help to see where I did wrong? There are 10 balls, and each ball to be place in bin 1 and bin 2. Each ball is placed indepedently. Let X be the number of balls in bin 1 and Y be the ...
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1answer
15 views

probability, transformation on Random Variable

This is a more general question about the transformation of a random variable. Say X is given as a certain distribution, and Y=g(X). If it asks to compute the pdf of Y, I am having trouble to ...
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3answers
36 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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0answers
45 views

probability, birthday paradox: need help to understand the solution

I need help to understand the following solution to a birthday paradox problem. problem:So you have $20$ people. Then let $P=$ # of pairs that share the common birthday. Compute ${\bf E}[P]$, ...
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2answers
42 views

Solving for a random number less than 401, generated by multiplying two numbers less than 21.

So, at a math meeting tonight we decided to play a game where you try to solve for a random number between 1 and 400 generated by multiplying two numbers between 1 and 20 together. Basically the ...
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2answers
53 views

Transformation on a random variable

Can someone please help me with formatting this question? $Y$ is an exponential random variable with parameter $1$. Let $Z=-Y$, what is the pdf of $Z$? Attempt: $$\Pr(-Y< y)=\Pr(Y>-y) ,$$ ...
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1answer
17 views

probability, depedence => uncorrelated?

One quick question. Two random variables are dependent, so it must be uncorrelated? Also for the terminalogy, does the term "uncorrelated" means two random variables have a corvariance factor of 0? ...
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1answer
22 views

probability, please help on bayes question

I dont need exact answer, I just need help to judge wether my following method is correct or not. Question:A physician has 5 patients. There are treatments A and B. Physician gives treament A to 3 ...
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1answer
21 views

probability, at least one is not present

you have $5$ red balls, $10$ green balls, and $15$ yellow balls in a balls. You randomly choose $5$ without replacement. What is the probability that at least one of the $3$ colors is not present ...
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1answer
24 views

What is the definition of sigma field generated by random variable $X$? [closed]

What is the definition of $\sigma$-field generated by a random variable $X$? And what does it mean?
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1answer
29 views

probability, need help on the marginal densities

I need help on the marginal densities. In particular, I know you just integrate the joint pdf f(x,y) from y=-infinity to +infinity, but in the context of the below question, I have trouble to define ...
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1answer
74 views

probabilty, random variable independent

Let $X$ and $Y$ be independent Poisson random random variables with ($\lambda=1$). Are $X-Y$ and $X+Y$ independent? Justify My attempt: $X-Y$ => random variable is $0$. $X+Y$=> Poisson of ...
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2answers
26 views

probability, indicator random variable

Let $A,B,C$ be independent events with $P(A)=P(B)=P(C)=\dfrac{1}{2}$. Let $X$ be the indicator r.v. of the event $A \cup B$ and $Y$ the indicator r.v. of the event $B \cup C$. Compute ${\bf E}[XY]$. ...
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0answers
6 views

What is the basis for why the linear congruential generator works as a good hash function?

The linear congruential generator (LCG) is often taught in introductory computer science classes as a good hash function. What is some mathematical justification for why the LCG works and why it works ...
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1answer
24 views

Generate some random matrix with given rank

Very often for creating new exercises (I teach basic matrix algebra), I need to a find a matrix $A$ such that: it has integer coefficients, not too big (in order to avoid big numbers computations) ...
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0answers
33 views

Help!!!!!!!!!!!!!! [closed]

1. Assume that books from a certain publisher have an average of one misprint every 20 pages. (a) What is the probability that a given page has two or more misprints? (b) What is the probability that ...
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2answers
34 views

probability: arithmetic with random variables.

I have a question of using arithmetic on random variables. Please refer to the following question, to which I will present my solution using the arithmetic (which I thought it's correct but actually ...
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1answer
20 views

Random points in sphere with probability p(r)

How to pick random uniformly distributed points in a sphere has been asked before. The difference is that I don't want uniform distribution, rather I would like the number density to scale by ...
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1answer
36 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
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0answers
16 views

Matlab function that returns samples of random process

Given an autocorrelation function for a random process $X(t)$, how could I write a Matlab function that returns samples of $X(t)$? (specifically $X(n\cdot \Delta t)$, $n=0,1...,N$ where ...
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0answers
6 views

Random numbers with groupings

I have 9 items, I need to pick a random item and I would like each item to be just as likely as the others to be chosen. I'm assuming my RNG is perfectly. Currently I am picking a random number ...
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1answer
54 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
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2answers
69 views

Is a number chosen at random necessarily irrational?

If I were to pick a number completely at random in the range [0,1), it seems to me that number would be irrational. After all, there are a countably infinite number of rationals between zero and one, ...
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1answer
28 views

Power spectral density of a stationary random process

The stationary random process X(t) has a power spectral density denoted by Sx(f). a. What is the psd of Y(t) = X(t) - X(t-T)? b. What is the psd of Z(t) = X'(t) - X(t)? What should the approach ...
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1answer
28 views

Random sample and probability

A random sample of 325 new toothbrushes showed that 14 were defective. What is the estimate of the probability that a new toothbrush is not defective? Either a toothbrush is defective or not. What is ...
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23answers
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Is an SD card a fair coin… to me?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
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0answers
25 views

Generate correlated random numbers precisely

Let's assume I want to generate k samples of n random numbers, that are correlated according to a given correlation matrix C (e.g. $n = 3$): ...
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0answers
17 views

Probability in a weighted Random

This is actually a question that came up whilst programming, but i figured this is more a question for the mathematicans: in a database have a set of roughly 500 entries. now i want to get 10 random ...
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0answers
19 views

testing hardware random number generators? [closed]

1)I would like to know how to test hardware random number generators. 2)What techniques ,tools or tricks to solve the problem ? 3)Any practical difficulties, implementation complexities etc
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1answer
29 views

Can 3 random variables have pairwise correlation -1?

Can 3 random variables X, Y, Z have pairwise correlation -1? This seems to be easy question, but I just got confused over this.
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0answers
17 views

Diameter of graph formed by pairwise intersections of random chords in a circle

Given n random chords in the circle (a random chord is chosen by specifying both endpoints uniformly randomly from the circumference), form a graph G on n nodes such that an edge exists between i and ...
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1answer
58 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
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1answer
34 views

Given the distribution of a random variable $R$, who do you get a uniform random variable $U$?

Let us say you have a random variable $R$. How would one generate a uniform random variable $U$, with the maximum possible entropy (or infinite entropy, if $R$ has such)? (For simplicity, you may ...
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1answer
70 views

sum of independent Rayleigh random variables [closed]

How do I find the probability density distribution (pdf) of the sum of independent Rayleigh random variables (whose probability density functions are known)? where is the reference? Could anybody ...
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1answer
20 views

Density on the square, expected value

Let $f: [0,1]^2 \rightarrow \Bbb R^{+}$ a density function on the square. I suppose that the random variable $X=(X_1,X_2)$ has the density f with respect to the lebesgue measure. I denote ...
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1answer
30 views

Moments of Geometric Random Variable

Let $X$ be a geometric random variable i.e. it represents the number of consecutive failures before you get the first success where the success probability is $\rho$. We know $E[X] = 1/\rho$ and ...
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21 views

Understanding graph obtained from Monte-Carlo simulations

I am running a Monte Carlo Simulation where I sample from about 65 Normal Distributions. I also keep track of the probability associated with each sample by approximating a thin area in the Normal ...
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1answer
33 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
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131 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
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2answers
34 views

How to create a random number generator that is completely random.

I'm trying to write an autonomous music composition program that makes use of many random number generators in order to make seemingly random melodies but I have no clue how to formulate a rigorous ...
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1answer
29 views

How to Create a Plane Inside A Cube

I have a $e \times e \times e$ cube and I want to create random planes with equation $ax + by + cz + d = 0$ inside this cube. I will put random points on those randomly created planes as well. Here ...
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2answers
43 views

Consider throwing two six-sided dice.

Let X be the sum of the two values and let Y be the product of the two values. What is the value of P(Y = i) for i = 1,2,3...36. I am having trouble approaching this problem. We are learning about ...
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1answer
85 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
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Random sample taken, what is probability?

It was determined that 22% of all stock investors are retired people. In addition, 38% of all U.S. adults invest in mutual funds. Suppose a random sample of 20 stock investors is taken. a. What is ...
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2answers
203 views

$e$ popping up in topic I'm unfamiliar with

I programmed up a little algorithm that goes like this: Fix two positive, real numbers, call them $\alpha$ and $\beta$. Generate a new, random, real number, $x \in [0,1]$ Set $\alpha$ = ...
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1answer
24 views

Probability of occurrence after Latin Hypercube sampling and then random sampling

I am using Latin Hypercube sampling to obtain numbers from a Normally Distributed set of data, so that I get a uniform spread of numbers across the Normal Distribution. I then select a number at ...
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1answer
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Probability of numbers within a Latin Hypercube

What is the probability of occurrence of numbers in a Latin Hypercube? If I have a 1 dimensional Latin Hypercube of 1000 numbers would the probability of each number just be 1/1000? Essentially, I am ...
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1answer
32 views

Generating Different types of Matrices in Matlab

I am working on a project for a numerical methods class comparing two iterative methods for solving $Ax=b$, and I was wondering what type of functions Matlab has for generating arbitrarily large ...
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0answers
53 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...