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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13
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366 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that ...
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 ...
3
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0answers
109 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
3
votes
0answers
48 views

What is the difference between these two questions?

Consider two independent random variables: X~U(0,1) and Y~U(0,2). Let Z = min(X,Y). b) Find F$_Z$(z) in terms of F$_X$(.) and F$_Y$(.). c) Eliminate F$_X$(.) and F$_Y$(.) to find F$_Z$(z). What is ...
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: ...
2
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0answers
33 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 ...
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 ...
2
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0answers
34 views

Inconsistent answers with conditional expectations

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1]$. I want to compute: $$E[X+Y\mid\max\{X,Y\}≤(1/2)]$$ My first approach was the following. Let $X=\max\{X,Y\}$. ...
2
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0answers
94 views

Question on uniform distribution of points on a sphere.

Let N points be uniformly distributed on the surface of a unit sphere $S^2$. What is the probability that every spherical cap of area A contains at least one point? The area $A$ depending on the ...
2
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0answers
115 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
2
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0answers
22 views

orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
2
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0answers
50 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
2
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0answers
58 views

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$.

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I ask this ...
2
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0answers
47 views

Expected value of winning with a Random Uniform Variable

I have this small homework I don't quite understand. The problem is formulated as a game. (Who wants to be millionaire) You start with 1£. money = 1.0£ You can choose to quit at anytime So you can ...
2
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0answers
96 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
2
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0answers
1k views

Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
2
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0answers
48 views

Is $e$ uniformly distributed in all bases?

There has been talk of whether or not $\pi$ is normal, i.e. uniformly distributed in all bases $b$ where $b\ge2$. The general response has been that we expect that it is, and have found no obvious ...
2
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0answers
269 views

Max variance of uniform distribution?

Suppose I roll a 20-sided die 1000 times and count the number of times a particular value comes up. This gives an array of 20 counts, and the expected value of each is 1000/20 = 50. I'd like to find ...
1
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0answers
12 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 ...
1
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0answers
21 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: ...
1
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0answers
47 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, ...
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0answers
17 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
1
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0answers
55 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
1
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0answers
46 views

Continuous Uniform Distribution

Suppose $X$ follows a continuous uniform distribution from 1 to 5. Determine the conditional probability $P(X > 2.5 | X \le 4)$ I am not sure I know how to do ...
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0answers
70 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...
1
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0answers
29 views

sum of random variables?

$x\to$ uniformly distributed on $(0,1)$ $y\to$ uniformly distributed on $(0,2)$ $z\to$ uniformly distributed on $(0,4)$ What is the probability that $2x+3y < z$? I tried to do it geometrically ...
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0answers
23 views

uniform distribution probability and mle

For part a), isn't the probability = 1? And I'm not sure what happens as $n\rightarrow\infty$; isn't the probability 1 also?
1
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0answers
43 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
1
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0answers
43 views

Obtain distribution of mid-range in uniform

I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation. $M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = ...
1
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0answers
14 views

Even dirstibution of a small set of random choices into a small set of buckets

Is there a way to evenly distribute randomly selected small set of items from a relatively larger set into to a small number of buckets using a hash function? For ex: Randomly select 20 numbers from ...
1
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0answers
30 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
1
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0answers
39 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
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0answers
29 views

Question about the uniform distribution and random variable operation

If I have a random variable with uniform distribution ranging from 0 to 1, as follows: $ X \sim U\left ( 0 ;1 \right ) $ And if I subtract that random variable $ X $ from 1, will the resulting random ...
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0answers
33 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
1
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0answers
273 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
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0answers
81 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
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0answers
313 views

pdf of an uniform distribution in matlab?

I'm reading a book and I came across a problem in which I should generate a uniform random variable and use hist, mean and std ...
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0answers
22 views

Injective uniform distribution on an n-sphere

I just asked this question on the stats stackexchange, but I thought that maybe someone on math knew the answer. So: For an application I'm working on, I need to go from some uniformly distributed ...
1
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0answers
28 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
1
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0answers
96 views

Second moment of random variable in the integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
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0answers
32 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
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0answers
23 views

Fast uniformity test within a ball.

Assume I have a dataset lies within a ball centered around the origin, I want to test the uniformity of the point distributed in the ball. In addition, I have all the distances to the origin computed ...
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0answers
54 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 ...
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0answers
88 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 ...
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0answers
40 views

choose a list of words such that have equal letter frequency

I have a big list meaning full Words. surely letter frequency of this word list is different for each letter. Now my problem is to find a way to randomly select words from this word list to a new ...
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0answers
666 views

What is the expected time you have to wait until the first bus comes?

 three buses, bus A, B, and C come to a bus stop every hour. The time at which each bus arrives at the stop is distributed as a uniform random variable, i.e., TA,TB,TC ∼ Unif[0,1] hours. The ...
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0answers
90 views

Uniform distribution

You arrive at a bus stop at 10'0 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. what is the probability that you wait longer than 10 minutes? if at ...
1
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0answers
101 views

Finding Limits of Integration

I have two functions, one depending on $x$ which is $\frac {1} {2} {\delta(x-5)} + \frac {1} {4}$ which is the combination of a dirac delta function at $5$ and a uniform distribution from $5$ to $7$. ...
1
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0answers
85 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...