For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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1answer
24 views

Box-Muller method to Polar Marsaglia scheme

I have just learned the Box-Muller method for generating normal random values. My notes then consider the Polar Marsaglia method, which is more efficient than Box-Muller. In Box-Muller: ...
0
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1answer
20 views

How do we know the pdf of $X_{max}$ for a uniform distribution? [closed]

How do we know the pdf of $X_{max}$? We have a sample of size $n=6$ from a uniform distribution of the interval $[0,\theta]$. Let $\hat{\theta}=X_{max}$ be the estimator of $\theta$. (a) Given ...
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0answers
33 views

Tukey's symmetrical lambda distribution

U ~ Uniform(0,1) $$Z_\lambda = \frac{U^\lambda-{(1-U)}^\lambda}{\lambda} $$ I have to find the first four moments and two ($\lambda_1,\lambda_2$) such that they have the same four moments.
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0answers
11 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
0
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1answer
28 views

What is the conditional distribution function of P(T < z| U=x) where U, V are uniformly distributed on [0,1] and T= Max(U,V)

Let U, V be independent random variables, both uniformly distributed on [0,1].T= max (U, V). 1) What is the joint distribution function of T and U, P(T <= Z, U <= x)? I did the following: P(T ...
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1answer
34 views

Understanding Bayes' theorem with uniform distribution

Station $X$ begin to transmit a message in $[0,20]$ with uniform distribution, and $Y$ also want to transmit a message in $[6,14]$ with uniform distribution. Assume that transmission takes $2$ ...
1
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1answer
23 views

Integrating PDF of Continuous Uniform RV to get CDF

For a continous uniform RV $$\text{PDF} = \frac{1}{b-a} \text{ for } x\in[a,b)$$ (sorry cant figure out how to add whitespace) and $$\text{CDF} = \int_{-\infty}^x \text{PDF} \, dx$$ so for $x>0$ ...
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0answers
26 views

Distribution of a function of Discrete Uniform RVs

Suppose $X$ and $Y$ take the values ${0,1,...,N}$ with probability $\frac{1}{N+1}$. What is the PMF of $|X-Y|$? I started it like so: Let $|X-Y| = Z$ $P(Z<z) = P(-z < X-Y<z)$ In the ...
2
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0answers
13 views

Targetting the distribution of distances between points

For a certain problem, I need to make distance dependent statistics, but with the constraint that the number of sampling points, $N$, should be kept as small as possible. To be more specific I need to ...
0
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1answer
46 views

Find the joint pdf of X and Y for two uniform distributions

Let X have a uniform distribution on the interval $(0,1)$. Given that X = x, let Y have a uniform distribution on the interval $(0,x+1)$. Find the joint pdf of X and Y. Sketch the region where ...
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0answers
20 views

evaluation of this integral arising in probability theory

Let $U_1,...,U_n$ be independent, uniformly distributed RVs on $[0,n]$. Consider the ordered statistics of these, $$U_{1:n} < ... < U_{n:n}$$ So $U_{1:n} = \min\{ U_1,...,U_n\}$ and so on. ...
3
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1answer
37 views

Average waiting times

I have the following exercise, which I would like to solve: Company A run buses between New York and Newark, their bus leaves New York every half an hour starting from 0:00, 24h a day. Company B also ...
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0answers
21 views

Uniformly Random Polynomial

Hypothesis: all the polynomials and values are defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number. My question is related to information security. Assume (in a protocol) ...
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0answers
8 views

Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
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0answers
28 views

Convolution of a Pareto and a Uniform distribuion

I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
3
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1answer
34 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...
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0answers
23 views

Working out probability distribution from given uniform random variable

I'm not able to understand how they got from probability U is less than e to the power of x to e^-x? shouldn't it be e^x instead because we are given in our notes that probability U that's less than ...
0
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1answer
18 views

does calculating the arithmetic mean of a serie of standar deviation measures make sense?

I'm trying to test if a hash function mantains a even distribution of values. My plan is to generate a set X of hashes and a integer Y symbolizing "slots" to see how many of the hashes maps to the ...
8
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0answers
72 views

Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that ...
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2answers
47 views

A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability B arrives before A?

As the title indicated, A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability that One day B arrives before A? I am bit confused whether to use Poisson (because arrival ...
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2answers
38 views

If $X_1,X_2,X_3$ are three different uniform random variables, calculate $E(X_1 - 2X_2 + X_3)$. [closed]

Suppose that a random variable $X_1$ is distributed uniform $[0,1]$, $X_2$ is distributed uniform $[0,2]$ and $X_3$ is distributed uniform $[0,3]$. Assume that they are all independent. a) ...
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0answers
25 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 ...
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2answers
46 views

If the area of a square is uniformly distributed, then find the expectation and variance of the side length.

I'm told that the area of a square (denoted $A$) is uniformly distributed across the interval $[15,20]$, I'm then asked to find the expectation and variance of the side length but I can't work out how ...
0
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2answers
72 views

Probability that the inpatient guy arrived first

So the question asks: Tom and Jerry set up a meeting at a restaurant. Each one of them, independently of the other, arrives at some random time between 9:00 pm and 10:00 pm (that is, the arrival ...
3
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2answers
42 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let ...
2
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1answer
33 views

Combine discrete uniform distributions to achieve a discrete uniform distribution of a larger range?

How can I effectively combine multiple discrete uniform distributions of a limited range to achieve a discrete uniform distribution of a large range? I.e. given $unif\{a,b\}$ generate $unif\{a,c\}, ...
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1answer
73 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
3
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1answer
34 views

Is modular multiplication under a prime modulus uniformly distributed?

Let's say that I have a prime, $p$, and an $m \in Z_p^*$. Then, I draw $a \leftarrow Z_p^*$ uniformly at random. Will $am \mod p$ be distributed uniformly over $Z_p^*$?
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0answers
35 views

Rewiting an integral in the case of a uniform distribution

I would like to rewrite this equation: \begin{equation} \widetilde{A}(\varphi^*) = \left[\frac{1}{1-G(\varphi^*)}\int_{\varphi^*}^{\infty} A(\varphi)^{1-\sigma}g(\varphi)d\varphi\right]^{1/(1-\sigma)} ...
0
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1answer
25 views

Uniformly Distributed Random Variable over an interval

Let $X$ be a random variable uniformly distributed over a nontrival interval $[c,d]$, and let $Y = aX+b$. For what choice of real constants $a$ and $b$ is $Y$ uniformly distributed over [0,1]? ...
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0answers
27 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

The Statement of the Problem: Let $ \{ U_i \}$ be a set (sequence?) of iid random variables such that $U_i \sim \text{Uniform}(0,1)$, and define $$ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt ...
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2answers
63 views

Probability that sum of independent uniform variables is less than 1

I would like to determine the probability $\mathbb{P}(X_1+\dots+X_n\leq 1)$, where $X=(X_i)_{1\leq i\leq n}$ is a family of independent uniform random variables on $[0,1]$. My first idea is to do this ...
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0answers
30 views

Probability function of the sum of multiple independent uniform distributions

Given n number of variables with uniform distribution. What is the probability distribution of the sum of these variables? Let's say that $a_1$ and $a_2$ are two independent uniform distributions in ...
0
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1answer
33 views

CDF and PDF of semaphore waiting time

Imagine we have a semaphore that alternates every 40 seconds between green and red. Waiting time is 0 when the semaphore is green, and when it is red it is the remaining time until it turns green. I ...
0
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2answers
53 views

What is the expected value of the sum of the $k$ (out of a set of $n$) smallest uniform random variables?

I know that the expected value of the sum of $n$ random variables is the sum of the expectation of each one. The expected value of a uniformly distributed random variable $U(a,b)$ is also well known ...
0
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1answer
21 views

subset of a uniform random number.

I need to create a 2-D random number generator that generates numbers in a convex region. Say, for example, this region falls within the area [-1,1]2. Would the following process lead to a uniform ...
0
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0answers
30 views

finding probability using uniform probability density

I am trying to solve: A value x is chosen with uniform probability density from the interval $[0, 1]$. Calculate the probability that $x < 1/2$ given that $x < 2/3$. I plotted these lines on ...
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3answers
54 views

pdf for random variable over unit disk

Let $(X,Y)$ be uniformly distributed on the unit disk $S = \{(x,y) \in \mathbb{R}^2:x^2 + y^2 \leq 1\}$. a) Find the probability density function for the RV $U = X + Y$. b) Find the probability ...
0
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1answer
23 views

CDF of the highest result of multiple unform random variables.

Say I have multiple uniform random variables. I want to know the CDF for selecting the highest result of all the variables. As an example, say I have 3 uniform random variables from [0, 100). Using a ...
0
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0answers
19 views

Distribution of maximum frequency in uniform sample

If I take $n$ random integers from $1$ to $m$, how do I calculate the distribution of the number of occurrences of the most frequent number? Any hints or initial approaches? I thought to get the ...
1
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1answer
44 views

Finding MLE of uniform distribution with actual example values

I'm watching this video and going to part I am stuck at here https://youtu.be/XaAtkCzdjLE?t=6m2s Following the example in the video, I assume that $\theta$ will be between $14$ and $501$. Now I ...
0
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1answer
49 views

Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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0answers
2k views

Mean distance of random points on a rectangular grid

I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate ...
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3answers
60 views

Joint probability uniform distribution

I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one: We agree ...
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2answers
79 views

Product of i.i.d. random variables uniformly distributed on $(-1,1)$ converges almost surely to $0$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.
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2answers
42 views

Find CDF of $Z:=\min\{X,Y\}$

Let be $X,Y$ independent random variables with $X\sim\mathcal{U}(\{0,1\})$ and $Y\sim\mathcal{U}(\{0,1,2\})$. Compute the CDF of $Z:=\min\{X,Y\}$. My idea: $$ \begin{align*} P(\min\{X,Y\}\leq x) ...
0
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1answer
41 views

Show that $P(X=2) = \frac{2}{3} - \frac{13}{27}$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U = p$ is Geometric with parameter $p$. Show that ...
2
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3answers
48 views

If $N_{\mid\Lambda = \lambda} \sim$ Poisson ($\lambda$) and $\Lambda \sim$ unif$(0,5)$, find the probability of zero occurring.

The number of storms in the upcoming rainy season is assumed to be Poisson distributed, but with a parameter $\Lambda$ that is also random and uniformly distributed on $(0,5)$. That is, $\Lambda \sim$ ...
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0answers
16 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
0
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1answer
68 views

Explain why $E(X)=1.65$ and $Var(X)=1.64$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U=p$ is Geometric with parameter $p$. (a) Find ...