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Questions tagged [uniform-distribution]

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

2
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1answer
34 views

$U_1,U_2,…$ i.i.d. $U[0,1]$, $P\sim Poi(\lambda)$, find $F_{\operatorname{min}(U_1,…,U_P)}$

Let $(U_n)_n$ a sequence of random variables i.i.d $U[0,1]$ and let $P\sim Poi(\lambda)$ a random variable such that $P$ is independent of $(U_n)_n$. Let $$ \\ X=\left\{\begin{matrix} \operatorname{...
0
votes
1answer
18 views

The pdf of sum of -log($U_i$) in which Ui is iid uniform distributed

Suppose $Ui$ is independently uniformed distributed between [0,b], $Y = -\Sigma_1^n log(U_i)$. what is the pdf of Y? I tried used characteristic function but it doesn't match each of usual ...
1
vote
1answer
19 views

Convergence of sequence of uniforms

Let $X\sim\mathrm{ Uniform}(0,1)$. Consider the sequence $X_n = X^n$. I want to study the convergence in law of this sequence. I did it using the distribution function, I have: $$F_X(x) = x \mathbb{...
0
votes
2answers
42 views

If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$

I'm stuck on this question, can someone help me, many thanks. If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$? update(this problem has been solved): I made a ...
-3
votes
1answer
46 views

$X_1,X_2,\ldots,X_n$ follows iid. $U(0,1)$. Then $\lim_{n\to\infty} P\left(\sum_{i=1}^nX_i \le \frac n{2}+n^{3/4}\right)$=? [on hold]

$X_1,X_2,\ldots,X_n$ follows iid. $U(0,1)$. Then $\displaystyle \lim_{n\to\infty} P\left(\sum_{i=1}^n X_i \le \frac n{2}+n^{3/4}\right)$=?
0
votes
1answer
21 views

find the probability about sum of random variables

Let $X_1, X_2, X_3, Y_1, Y_2, Y_3, Z_1, Z_2, Z_3$ be random variables which have uniform distribution between 0 and 1. It means, the average of $X_1 = 0.5$ Let $X=X_1 + X_2 + X_3$ and $Y=Y_1 + Y_2 + ...
-1
votes
0answers
24 views

Show that $\frac{4(X_1+\cdots + X_n)-n^2}{n^{3/2}}$ converges in distribution

Let $X_n$, $n\geq 1$, be independent and identically distributed uniform on $[0,1]$ random variables. Prove that $\frac{4(X_1+\cdots + X_n)-n^2}{n^{3/2}}$ converges in distribution and identify its ...
0
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0answers
39 views

Let $M$ ~ $Geometric(p)$ and $X \vert M = m$ ~ Uniform on integers 1…m

Let $M$ ~ $Geometric(p)$ and $X _{\vert M = m}$ ~ Uniform on integers 1....m. I want to find the density of X and its expected value. This is what I have done so far, but I am not sure it is right: ...
1
vote
1answer
21 views

Uniform and exponential distribution

Consider an experiment. The duration of the experiment has uniform distribution on $[2,6]$h. When the experiment starts, the device A turns on. This device will turn off after time,which has ...
-2
votes
1answer
38 views

Let $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(\frac{X_1+X_2}{X_1+X_2+X_3}$)? [closed]

Let $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(\frac{X_1+X_2}{X_1+X_2+X_3}$) ?
1
vote
1answer
32 views

Hints on proving existence of $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$

Let $(X_{n})_{n}$ be independent random variables that are $\mathcal{U}{[1,2]}$ Prove $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$ exists for $n \to \infty$ and that $\exists c \in \mathbb R$ such that $(...
0
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0answers
13 views

Determine uniform parameters in mixed Poisson model

I have to determine the continious uniform parameters in a certain Poisson distribution. To make it a bit clearer: $X_i $ ~ $Pois ( \Lambda) $ and $\Lambda $ ~ $Unif (a,b)$ where $a = 0$. Also ...
1
vote
0answers
53 views

“Fragmentation” of a distribution (from paper)

I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to ...
0
votes
1answer
39 views

uniform distribution on [0,1] find function

Consider $X∼unif [0,1]$. Find a function $g: \mathbb{R} \longrightarrow \mathbb{R}$, such that g(X) has pdf $f(t) = \begin{cases} {t+1}, & \text{$-1 \leq t\leq 0$} \\ {1-t}, & \text{$0<t\...
3
votes
1answer
45 views

$\ X_i $ is discrete random variable, Compute $\ \sum X_i = 97 $

Let $\ X_1, X_2, , \dots , X_{10} $ be a discrete random variable with uniform distribution between $\ 0 $ to $\ 10 $. Compute $\ P\{ \sum_{i=1}^{10} \ X_i = 97 \} $, the variables are independent. ...
1
vote
0answers
36 views

Find $P(\min\{n\gt0:X_1+X_2+\cdots+X_n\gt x\}>2)$ for $(X_n)$ i.i.d. uniform on $(0,1)$

Let $X_i$ be i.i.d random variables that follow uniform distributions on $[0,1]$. Let $t(x)=\min\{n\gt0:X_1+X_2+...+X_n\gt x\}$. Find $P(t(x)\gt2)$. Here's my answer but I'm not sure if I interpreted ...
0
votes
1answer
42 views

Find probability that the equation $x^2+Bx+C=0$ has 2 distinct roots.

Let $B,C$ independent random variables such that $B\sim \operatorname{exp}(\lambda),C\sim U[0,1]$. I have 2 questions about the solution: "We're looking for the probability that $\mathbb{P}(4B^2-4C&...
1
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0answers
28 views

Binomial distribution/conjugate posterior

Let's consider some experiment with tossing a coin. NOTE: my question is given at the very last paragraph. Observation $y=0$ or $y=1$ [tails (T) or heads (H)], $p \in [0, 1]$ (probability of heads) ...
0
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1answer
22 views

How is the subtraction of a uniform (0, k) and its entire part distributed?

Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x. I think that [X] ...
1
vote
1answer
25 views

How to sample uniformly from ten urns with ten balls each

I have ten numbered urns with numbered 10 balls in each. I want to draw $n<100$ balls in a uniform distribution from all $100$ balls (the urns and all balls are distinct.) My procedure: I roll a 10-...
0
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0answers
19 views

probability of renewal process with uniformly inter arrival times

What is the probability the (N(t)=n) for a renewal process with interarrival time uniformly distributed on (0,1)? I thought that P(N(t)=n)== F (n)(t) − F (n+1)(t) where F (n)(t)=(t over n)/(n!). Do ...
1
vote
0answers
33 views

What is the Probability density function of $X^2$ where X is an Uniform distribution

I'm a student and I'm studying random variables and very new to it. I was studying the Uniform distribution and in it, it calculates the Expected of $X^2$ by $$ E\left(X^2\right) = \int_{- \infty}^\...
2
votes
2answers
23 views

product of quadratic forms of random vectors uniform on the sphere

Let $g = (g_1, ..., g_n)$ be a random vector distributed uniformly on the sphere $\{ x \in \mathbb{R}^n : \| x \|_2 = 1 \}$. Let $A, B$ be two symmetric $n \times n$ matrices. I am interested in a ...
0
votes
2answers
27 views

If $X\sim U(0,n)$, how can I show that $X-[X]\sim U(0,1)$?

If $X\sim U(0,n) ; n \in \mathbb N$ , how can I show that the distribution of $Y =X-[X]$ is $U(0,1)$? Any hint will also help me...
0
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0answers
15 views

Standard error of the variance of Uniformly distributed data

I would like to derive the standard error of the variance for a sample taken from a Uniform distribution with bounds $a$ and $b$. The standard error of the mean of that sample is straigtforward: $SE_{...
-1
votes
3answers
32 views

Are uniformly distributed random variables $X$, $X^2$ (in)dependent?

We have a random variable $X \sim U[-1,1]$. I should find out whether $X, X^2$ are independent or not. I have shown that $\textrm{cov}(X,Y) = 0$ and that PDFs are $f_X(x) = \frac{1}{2} $ for $x \in [-...
1
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0answers
32 views

Calculate the distribution of $(\cos(\min(X,Y)),\max(X,Y))$

Let $X,Y$ be iid distributed uniformly on $[0,1]$, my task involves calculating the distribution of $(\cos(\min(X,Y)),\max(X,Y))$ I have already calculated the distribution of $(\min(X,Y),\max(X,Y))$, ...
-1
votes
1answer
42 views

Calculate $E(X^2)$ with X uniformly distributed on [-1,2] [closed]

X is uniformly distributed on $[-1,2]$. Hence the density function should be $\frac13$ and E(X) = $0.5$. Now I want to calculate $E[X²]$. But to do that I need the density function of $X²$. This ...
0
votes
1answer
24 views

Compute $\ E[|X^2-16|] $ where $\ X \sim U $

$\ X \sim U (-4,7) $ with $U$ a continuous uniform distribution. X is continuous variable. Compute $\ E[|X^2 - 16|] $ So: $$\ E[|X^2 - 16|] = \begin{cases} E[X^2 - 16] , \ \ 4< X \\ E[-X^2+16]...
1
vote
2answers
16 views

Calculating mutual events with continuous distribution

Suppose $\ X \sim U(-4,7) $ Compute: $\ P\{X^2 - 16 > 0 \ | \ X > 0 \} $ My attempt: $$\ P\{ X^2 - 16 > 0 | X>0 \} = P \{ X > 4 | X > 0 \} = \frac{P\{X>4\}}{P\{X>0\} } $$ I'...
0
votes
1answer
27 views

two people meeting, uniform distribution

Two people, A and B meet in a bar between 6 pm and 7 pm. They arrive independently of each other, evenly distributed between 6 pm and 7 pm. The first person who arrived, do not wait more than 15 ...
3
votes
3answers
75 views

What is the probability $X + Y > 100$? $X,Y \sim U[40,80]$

I'm struggling to write out the integral for this. Both $x$ and $y$ are i.i.d. distributed uniformly over $[40,80]$. I think it should be of the form $\int_{40}^{80} \int_{100-x}^{80} \frac{1}{40\...
1
vote
1answer
59 views

MLE of $\theta$ in $U[0,\theta]$ distribution where the parameter $\theta$ is discrete

Consider i.i.d random variables $X_1,X_2,\ldots,X_n$ having the $U[0,\theta]$ distribution: $$f_{\theta}(x)=\frac{\mathbf1_{[0,\theta]}(x)}{\theta}$$ , where the unknown parameter $\theta\in\{1,2,\...
0
votes
1answer
18 views

Joint uniform distribution two different intervals

I have a problem with the following exercise: X corresponds to the duration of Paul’s commute to work, and Y to the duration of Peter’s. X is uniformly distributed in [15, 25] and Y in [15, 30]. X ...
2
votes
0answers
25 views

Normal approximation of sum of uniform independent RVs using CLT

Let $X_1$, $X_2$, ... $X_{16}$ and $Y_1$, $Y_2$, ... $Y_{16}$ be independent uniform random variables over the interval [-1,1] and let: $$ W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16} ...
0
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0answers
17 views

Transforming probability function ,expected value

Given the following problem where I have to calculate the expected value of a given $T(x) = \frac{x^2}{2}\le8$ function. During the class we converted the distribution into uniform distribution so ...
0
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0answers
10 views

exponential family distribution are main source to generate distribution characteristics

my question is that why we say this distribution is from exponential family and by comparing it with exponential form we find cummulant function ,sufficient statistics and more things, distributions ...
0
votes
1answer
33 views

Probability density function of $X$ with a uniform distribution on a unit sphere

Given that $f_\Phi(\phi)=\dfrac{\sin(\phi)}{2}$ for $\Phi\in[0,\pi]$, and $f_\Theta(\theta)=\dfrac{1}{2\pi}$ for $\Theta\in[0,2\pi]$, where $\Phi$ and $\Theta$ are independent. What is the PDF of $X=\...
0
votes
1answer
40 views

joint PDF of max and min of $n$ iid standard uniform random variables

Let $U_1, ... U_n$ be iid standard uniform variables. Let $X = max(U_i)$ and $Y = min(U_i)$. The goal is to compute the joint PDF of $X, Y$! I have already computed the PDFs of $X$ and $Y$ separately....
0
votes
1answer
11 views

Poisson Process expectation of time of an event given number of events until that time shows Uniform distribution characteristics

Question: On a weekday, buses arrive to a certain stop with respect to a Poisson process with rate $\lambda = 2$ per hour. A regular weekday is assumed to start at 6:00 AM. Given that exactly $10$ ...
0
votes
1answer
23 views

fint joint and marginal distributions of two uniformy distributed variables over a specified region

This exercise (partially) comes from Rice 3.9 Suppose that (X, Y ) is uniformly distributed over the region defined by $0 ≤ y ≤ 1 − x^2$ and $−1 ≤ x ≤ 1$. Find the marginal densities of X ...
1
vote
1answer
18 views

Find the marginal distribution of an point randomly chosen on an ellipse

This exercise comes from rice 3.6 and states: A point is chosen randomly in the interior of an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Find the marginal densities of the $x$ and $y$ ...
0
votes
1answer
24 views

joint PDF of 2 dependent variables w/ convolution

Let $X, Y$ be iid standard uniform variables, and let $T = X + Y$. The goal is to find the joint PDF of $X$ and $T$. The work I've done so far is to find the PDF of $T$ by evaluating the convolution, ...
0
votes
1answer
21 views

Convolutions: Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X.

Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X. Solution: $f_T(t) = \int_{-\infty}^{\infty}f_Y(t-x)*f_X(x)dx$ $= \int_{-\infty}^{\infty}1*\lambda e^{-\lambda x}dx$ Integrate ...
0
votes
0answers
16 views

Compute likelihood from a uniform distribution with known $\theta$

I'm having a hard time trying to understand how to compute the likelihood from a uniform distribution with pair variables ($X_1$, $Y_1$) ...) i.i.d when $\theta$ is known. Most examples I've found is ...
1
vote
1answer
12 views

Find PMF of X^2 if X~Dunif(0,1,…,n)

Follow up on this: Find PMF of $X^2$ if $X$~Dunif (I do not have enough "reputation points" to comment, so if this is an inappropriate way to ask for a follow up, please let me know) Is this a ...
1
vote
1answer
22 views

Let $U \sim \textrm{Unif}(0, \pi/ 2)$. Find the PDF of $\sin(U)$.

This is almost the same as Suppose that X ∼ U ( $− π/2$ , $π/2$ ) . Find the pdf of Y = tan(X)., but making sure I am understanding the process: Let $U \sim \textrm{Unif}(0, \pi/ 2)$. Find the PDF ...
0
votes
1answer
34 views

Proof or relation between a Uniform and Exponential

Given $X\sim U(0,1)$, i have to determine the density of $Y=-\frac{1}{\lambda}lnx$. I can't apply the law of transformation of random variables because $g(X)$ is not a monotonic function. So, i ...
0
votes
0answers
17 views

Sum of “n” independent random variables. [duplicate]

Let $X_1,X_2,...,X_n \sim U_{[0,1]}$. $X_1,..,X_n$ are independent. Let us define : $Y=X_1+X_2+...+X_n$ How to find the PDF of $Y$?
0
votes
1answer
13 views

Find the expectation of vertices

Question: A point is chosen uniformly at random from the triangle in the plane with vertices (0,0), (1,0), (0,2). Let X be the x-coordinate of this point, and let Y be the y-coordinate of this point. ...