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
17 views

Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
0
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0answers
12 views

Monotonicity of this conditional probabilty

$U_i ( i=1,...,n)$ are i.i.d and (0,1) uniformly distributed variables. Define its truncated product at threshold $\tau$ $(\tau\in(0,1))$ as $U=\Pi_{i=1}^nU_i^{I(U_i< \tau)}$. Define ...
2
votes
1answer
21 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
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1answer
30 views

uniform distribution, probability

This is regarding the previously asked question: Let $T_1$ and $T_2$ be random times for a company to complete two steps in a certain process. Say $T_1$ and $T_2$ are measured in days and they have ...
2
votes
1answer
23 views

Explanation for “jointly pdf is constant but marginal pdf is not”

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$ From short computation, we know: Jointly pdf: $f_{XY}(x,y) = 2$ Marginal pdf of $x$: $f_{X}(x) =\int_0^x ...
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0answers
11 views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
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0answers
20 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: ...
0
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1answer
28 views

Expected value of sum of uniformly distributed variables

Let $X_i$ be a uniformly distributed random variable on the interval $[-0.5, 0.5]$ that is: $X_i$ ~ $U(-0.5, 0.5)$, for $i \in [1, 1500]$ How can I calculate the expected value of the sum ...
2
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3answers
50 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
1
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1answer
26 views

Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$. I know that maybe by using Characteristic function of $S_n$ ...
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2answers
44 views

Probability Uniform Distribution Set Up Integral

Consider a $1$ meter stick and suppose you break it into two pieces $X$ meters from the end, where $X \sim \operatorname{Unif}(0,1)$. What is the expected length of the longer piece (after it is ...
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0answers
46 views

Probability and Uniform distribution lottery question

Suppose that a person has a lottery ticket from which she will win $X$ dollars, where $X \sim\mathrm{ Unif} (0,4)$. Suppose her utility function is $U(x) = x\alpha$ for $x \geq 0$ and $0$ otherwise, ...
1
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2answers
45 views

Uniform distribution and expectation

Let $U \sim \mathrm{Unif}(0,1)$, $X=U^2$ and $Y=e^X$. Compute $E[Y]$ (leave answer as an integral). So essentially we need to compute $E[e^{U^2}]$? I am a little confused how to approach this problem? ...
2
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0answers
26 views

Departure from uniformity in a continuous (time) distribution

I know how to quantify the departure from uniformity ( or a uniform distribution) for discrete distributions. Assume you have a distribution set of P: ...
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votes
1answer
24 views

Maximum of uniform random variable [closed]

Suppose I want to compute: $\max \{E[2 \min(\theta, I)]-I\}$ where $I$ is the choice variable and $\theta$ is a uniform variable in [0,1]. How should I do it?
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1answer
46 views

Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
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0answers
25 views

Conditional expectation of discrete uniform random variables with one fixed

Came across a problem that I worked on sometime ago having the following structure: Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...
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0answers
40 views

understanding uniform distribution on multiset

If I have a set of words that form a text, say text $a$, and a set of texts, I then calculate the similarity between text $a$ and each text in the set. Then, I get a multiset. For example: $$s = ...
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1answer
86 views

what is the probabilty that sum of two random numbers between A and B is less than third number C [closed]

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$? I have tried answering this question using graph method to find the area under the ...
3
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0answers
28 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
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0answers
29 views

Converting normally distributed numbers to uniform distribution

I have a Perlin noise algorithm I've written my self. It seems to produce gausian numbers at the range of -1.5 and 1.5 but I'll convert them to the range of -1 and 1. I' currently working on a project ...
0
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0answers
30 views

Product of n uniformly distributed RVs

Let $X_j \sim U(a,b)$. What is the PDF of $\prod_{j=1}^n X_j$? I have seen some with $X_j \sim U(0,1)$ but I was wondering what the general form of the solution is for any $a$ and $b$.
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5answers
48 views

PDF of function of uniform random variable [closed]

Why PDF of $g(X)=X^3$ is not uniformly distributed, when X is uniform random variable between $(0,1)$? As for every value of X there is unique value of $g(X)$, hence the probability density of $g(X)$ ...
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3answers
34 views

If $X$~$U(0,1)$, and $Y=2x-4$. What is the density function of Y?

If $X$ is uniformly distributed $\mathcal{U}(0,1)$ , then what is the distribute density function of $Y$? I thought that if $$fx(x) = 1/(1-0), \; \mbox{for} \; 0<x<1$$ then ...
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0answers
16 views

Understanding compound distribution and plotting the mixed distribution graph

If $N$ takes the values $0, 1$ and $2$ with probabilities $½, ¼ $ and $¼ $ respectively, and the $X_i$ ’s have a $U(0,10)$ distribution, draw a sketch of the frequency distribution of $S$. $N$ is the ...
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1answer
16 views

Finding a uniform distribution on the output of a multivariable function

Suppose we have a non-invertible continuous function that maps from some continuous interval ${I}^n$ to $\mathbb{R}$ with $n \ge 1$. To take an example, let $f(a,b,c) = a \cdot e^{-bc} - b \cdot ...
3
votes
2answers
70 views

A single, good test for a random number generator?

I'm chasing a bug in the RNG for a well-known programming language under certain pathological inputs. There is an obvious pattern in this pathological case, apparent with very small n (~ 10000), and ...
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0answers
25 views

How to reduce the standard deviation from $P$ where $P$ involves a random integer uniformly distributed in $\left[ 0,100\right]$

I have a probability $P$ derived from: - A random integer $A$ uniformly distributed on its range such that $A\in\left[0, 100\right]$ - An integer $K$ such that $K\in\Bbb N$ - A number $X$ such that ...
1
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1answer
23 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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2answers
64 views

Why birthday distribution is not uniform. [closed]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
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3answers
36 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
3
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1answer
51 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
2
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1answer
50 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
2
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0answers
32 views

Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
2
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0answers
39 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
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2answers
44 views

Uniform distribution on $\{1,\dots,7\}$ from rolling a die [duplicate]

This was a job interview question someone asked about on Reddit. (Link) How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die? Presumably, you are to do this by combining ...
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1answer
29 views

Uniform distribution on sphere

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.
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1answer
50 views

Distribution of the difference of order statistics

Let $X_1$ and $X_2$ be a random sample of size 2 from a uniform distribution over the interval $[0,1]$, let $Y_1$ and $Y_2$ be the corresponding order statistics. Find the conditional density ...
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2answers
64 views

The probability density of $X^2$?

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? ...
0
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1answer
27 views

Calculate $P(|X-4| > 1.5)$

If $X \sim U(2, 8)$ Would it be $$P(X > 1.5 + 4) + P(X <-1.5 +4)$$
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0answers
25 views

Exlain the significance of the uniform random variable for the simulation of random variables

I can think of the "Universality of the Uniform": Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts ...
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1answer
19 views

unbiased estimator of $m^2$ of uniform distribution over $(0,m)$ [closed]

I have a sample of size $1$ from a distribution that has $U\left(4,4+m\right)$ as probability density function. Is there any unbiased estimator for $m^{2}$?
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0answers
30 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
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votes
1answer
45 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
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1answer
53 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
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1answer
69 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
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0answers
35 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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2answers
125 views

Average distance from center of circle to evenly-distributed points within it

With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a ...
0
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0answers
45 views

Expectation of $v=\inf \{n\geq 2\,;\, X_n > X_1 \}$ when $(X_n)$ is i.i.d. uniform on (0,1)

Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...
2
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2answers
47 views

Maximum likelihood estimator on uniform distribution

I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me mad. Here it goes. Consider the uniform distribution on ...