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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Difference Uniform rv's

Let $U_{1}\sim U(0,1)$ be a standard uniform random variable. Is $U_{1}-U_{1}$ uniformly distributed? I've been trying to work this out as follows: Let $A,B$ be rv's $$P(A-B\leq ...
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0answers
77 views

Uniform Distribution Probability

A manager of an apartment store reports that the time a customer on the second floor must wait after calling the elevator has uniform distribution ranging between 0 and 6 minutes. Assuming that it ...
0
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2answers
43 views

Finding the probability density function $Y=|X|$

Suppose $X$ is a uniform $([-1,2])$ random variable. How can I find the probability density function $Y=|X|$?
3
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2answers
35 views

Independent two random variables with uniform density

Let $X \sim U[-1,1]$ and $Y=X^2$. Show that $X$ and $Y$ aren't independent. Of course we have $F_X(x) = \frac{x+1}{2}$ and $F_Y(y) = \frac{\sqrt{y}+1}{2}$. But how can I find $$F_{XY}(x,y) = \Pr(X ...
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0answers
122 views

Geometric Mean of Uniform random variables convergence

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
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0answers
26 views

low discrepancy of halton sequences

I want to proof, that the Halton sequence is low discrepancy. I have to show that $$D_N^*(\mathcal{S})\le C\frac{\ln(N)^s}{N}$$ where $D_N^*$ ist the star discrepancy, $\mathcal{S}$ is the Halton ...
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1answer
79 views

Expected value with two random variables

A line segment AB of length 1m is broken in two at a random point P where the length of AP has the following probability density function: $f(x)=6x(1-x), 0<x<1$ A point Q is uniformly selected ...
0
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1answer
124 views

Complete Statistic: Uniform distribution

Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le \theta$. I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete. Now, I know I should ...
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3answers
47 views

How do I calculate this expected value?

The problem is as follows: Six players draw, one after another and independently, a number uniformly distributed on $[0,1]$. A player is called a recordist if he draws a number that is larger than ...
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1answer
38 views

Moment of uniform distribution

Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$. Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ ...
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2answers
260 views

Uniform Distribution in [0,1] where P[x1+x2<=x3]

Consider the following question : X1, X2, X3 are 3 independent random variables having uniform distribution between [0,1] then P[x1+x2<=x3] to the greatest value is ? Now this is not a homework. ...
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1answer
33 views

Proving a process generates a uniform distribution

I have a process that generates a series of real numbers. Specifically, starting from a given arbitary value (Xi-1), the process will generate a new number Xi following the formula: Xi = Xi-1 + ...
2
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2answers
112 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
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1answer
32 views

If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability X value is greater than Y value

Suppose $X$ and $Y$ are continuous uniform random variables. If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability that a random $X$ value is greater than a random $Y$ value.
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0answers
48 views

Choosing points uniformly on a sphere surface

I need a set, $A$, of points on a sphere surface, $S$. $A$ must satisfy: 1. The mean is the exact center of the sphere. 2. $\forall p_1,p_2\in S:\sum _{i\in A} \text{od}\left(i,p_1\right)=\sum _{i\in ...
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0answers
109 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
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1answer
105 views

Sum of three numbers from unformly distributed set equals to zero

I'm reading Sedgewick's "Algorithms" and completely stuck at one exercise. It is formulated like that: Develop an appropriate mathematical model describing the number of triples of N random int ...
3
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1answer
24 views

CDF on Standard uniform gives the same distribution

Assume that $X$ has a continuous and strictly increasing CDF $F_X$. Define $Y = F_X^{-1}(U)$ where $U$ is standard Uniform. How dow I show that $X$ and $Y$ have the same distribution?
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1answer
23 views

Computing an expectation with a uniform probability distribution.

Suppose $F$ is a cumulative density function of a uniform distribution between $a=0$ and $b=B$ and $c$ is a positive real number. I need to evaluate the integral $$\int_c^Bq\;\mathrm dF$$ where the ...
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1answer
40 views

Uniformed Distribution - Recap

I have divide the interval $[0,1]$ into $k$ equal sub-intervals, which I call classes, and generated $n$ observations from a uniform distribution. The number $X_{1}$ of the $n$ observations that fall ...
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0answers
55 views

What is the conditional distribution of this random vector?

Let us have random vectors $X_1, \dots, X_N$ which are identically independently uniformly distributed in the $n$-dimensional unit hyperbox $[0; 1]^n$. Let $c = (0.5, \dots, 0.5)$ be the center of ...
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1answer
34 views

Method of moments on uniform distributions

I need help on how to find the estimates $a$ and $b$ in the uniform distribution $\mathcal U[a,b]$ using the method of moments. This is where I am at: I have found $U_1=\overline X$ and ...
0
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2answers
53 views

Distribute a population size based on fractions using random number generator drand48()

I have a population size of say 5000 people. Every person belongs to either A, B, C or D category. I want to split the population as per a given fraction provided by user. for example, 99% of A, 0.4% ...
2
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2answers
122 views

$X$ and $Y$ are uniformly ditributed on $(0,1)$. distribution of $\max(X,Y)/\min(X,Y)$

Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution ...
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2answers
227 views

Transformation of a uniform distribution in order to get a random variable distributed like Y.

$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$. is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known constant and ...
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2answers
74 views

Calculate expectation and variance

Let $(X_n)$ be a sequence of independent RVs which are uniformly distributed on $[0,1]$ interval. For $0<x\le 1$ we define $$N(x):=\inf\{n:X_1+\dots+X_n\ge x\}.$$ Show that $$\mathbb{P}(N(x)\ge ...
2
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1answer
64 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
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0answers
114 views

Expected Value - Uniform distribution over infinite interval

Question: The probability that an error is introduced into a packet is $\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of ...
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2answers
65 views

Probability with multiple uniform distributions

Question: Two sources output a number at equal rates. The output from source A is uniformly distributed between 100 and 199, and the output from source B is uniformly distributed between 50 and 249. ...
2
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2answers
75 views

Calc expected value of 5 random number with uniform distribution

Assume we have a random numbers $\sim U(0,100)$. Then the expected value of that number will be: $\int_{0}^{100} \frac{x}{100}$ = 50.5 Now assume we have 5 random numbers $\sim U(0,100)$. How can I ...
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3answers
115 views

uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as Mx(t) = ((e^(tb)-e^(ta))/t(b-a) However I know Mx(0)=1 but I can't get my head around how this is possible as if t=0, ...
2
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1answer
606 views

Probability the three points on a circle will be on the same semi-circle

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ ...
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0answers
47 views

Prove that $E(X_1 \dots X_n)^\frac{1}{n} \leq (EX_1 \dots EX_n)^\frac{1}{n}$, with $X_i$ uniform distribution

Let $X_1, \dots , X_n$ i.i.d. uniformly distributed random variables with $f(x) = 1_{(0,1)}(x)$, $x \in \mathbb{R}$. Let $\Pi_n = (X_1 \dots X_n)^\frac{1}{n}$ and $M_n = \max \{ X_1, \dots , X_n ...
2
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1answer
42 views

Finding the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$

I'm trying to find the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$ Here $U$ is the uniform distribution. The method I use i to introduce an auxilary variable $W=X$ and then use ...
3
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1answer
196 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
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2answers
72 views

How to show $\max\{Y_{1},Y_{2},\cdots,Y_{n}\}$ converges in probability to $\theta$ as $n \to \infty$.

Let $Y_{1},Y_{2},\ldots,Y_{n} $ be independent random variables , each uniformly distributed over the interval $(0,\theta)$. Show that $\max\{Y_{1},Y_{2},\ldots,Y_{n}\}$ converges in probability to ...
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1answer
34 views

Simple probability problem.

$X$ ~ $U(0,1)$ and $Y$~U$(0,1)$ are two indenpendent variables. Get Pr ( Y > X). NOW what i don't understant in this problem is how you set the limits of integration. I heard that you must set ...
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1answer
228 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
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1answer
83 views

Finding density function for uniform distribution

Can anyone help me set this up correctly, please: John is going to eat at at McDonald's. The time of his arrival is uniformly distributed between 6PM and 7PM and it takes him 15 minutes to eat. Mary ...
2
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1answer
298 views

Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, ...
0
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1answer
375 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
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1answer
146 views

Density of uniformly chosen random point inside triangle

Imagine the triangle inside of the points $(0,0), (0,1)$ and $(1,0).$ Let $(X,Y)$ be a uniformly chosen random point from the triangle. Then find the joint density of $X$ and $Y$. The answer is ...
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1answer
39 views

Weak convergence and limiting distribution

I have $X_{i} \sim \operatorname{Unif}\left(0,1\right)$ iid random variables and have to show that $$ \frac{4\sum_{i=1}^n iX_{i} - n^2}{n^{3/2}}$$ converges weakly and compute its limit. How can I do ...
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2answers
76 views

Find the probability of $ x_2/x_3 \leq a $ where $x_2,x_3$ are uniform i.i.d.

Let $x_1,x_2,...,x_n $ be independent and identically distributed, uniformly on $(0,1)$. How to find $P(x_2/x_3 \leq a)$?
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1answer
214 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n ...
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1answer
77 views

Bivariate and Multivariate Probability Distributions

For my homework for Bivariate and Multivariate Probability Distributions section, I encounter the terms joint density, joint distributed random variable, joint probability, uniform distribution, when ...
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2answers
303 views

Uniform distribution over the unit circle

Suppose that $U$ and $V$ are two independent uniform $(-1,1)$ random variables. Any hints on how I can show that their conditional distribution, given $U^2 +V^2<1$ is given by the uniform ...
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1answer
65 views

$(\log n)_{n \in \mathbf{N}}$ not uniformly distributed mod 1.

Let $(x_n)_{n \in \mathbf{N}}$ be a sequence of real numbers, we say that $(x_n)$ is uniformly distributed mod 1 (u.d. mod 1) if $$\lim_{N \to \infty} \frac{|\{1\leq n \leq N : (x_n -\lfloor x_n ...
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0answers
68 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
2
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1answer
35 views

Probability that at least one event (out of two uniform RV) happens before two other random events

I recently faced a probability problem that is puzzling me. I would like to ask you if you could help me. I have two random variables X1 and X2 i.i.d with uniform distribution U[64,96] and other two ...