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.

Filter by
Sorted by
Tagged with
1
vote
2answers
36 views

MLE for uniform distribution around $[-\theta,\theta]$ [duplicate]

Given $X_1,\ldots,X_n$, where $X_i\sim U(-\theta,\theta)$, what the MLE for $\theta$? Apparently the answer is $\max\{|X_1|,\dots,|X_n|\}$ but I can't figure out why. The density function is $$f(x,\...
0
votes
1answer
27 views

Distribution - Line segment quotient

Visualization of the problem Consider a line of the interval $[0,2]$ that gets divided into two parts by randomly (according to $Uniform([0,1])$ choosing one point $w$ of the interval $\Omega :=...
0
votes
1answer
29 views

Similar probabilistic results for 2 different (discrete) random variables

I have a general question, but I will motivate it with an example. Let $X_s$ be a random variable in $\lbrace1,2,\ldots\rbrace$ that follows the Zeta distribution $\zeta(s):$ $$\mathbb{P}(X_s=k)=\...
0
votes
0answers
22 views

Minimal Sufficient Statistic for $U(0, \theta)$

The definition of a Minimal Sufficient Statistic (MSS) denoted $S(X)$ is $$ \frac{L(\theta;x)}{L(\theta;x)} \text{ independent of $\theta$} \iff S(X) = S(Y), $$ assuming the densities exist and $L$ ...
0
votes
2answers
30 views

Finding CDF and PDF of $Y=20/X$ when $X$ is uniform on $[4,7]$

I have a problem where $X$ is uniform on the interval $[4,7]$ and $Y = 20/X$. I am asked to find $F_Y(y)$ and $f_Y(y)$ using the CDF and PDF. This is a uniform distribution, so it's easy enough ...
4
votes
1answer
114 views
+250

Uniformly Distributed Random-Variable With Specific Ordering

Let $0\leq a<b$. Define the subset of $[a,b]^n$ by $$ X=\{(x_1,\cdots,x_{n-1})\mid b^{2n}\geq x_1^{2(n-1)}\geq\cdots\geq x_{n-1}^2\geq a\} $$ What is the probability that a uniformly distributed ...
1
vote
1answer
16 views

Probability (Uniform Distribution Question)

Question: A large wooden floor is laid with strips 2 inches wide with negligible space between the strips. A uniform circular disk of diameter 2.25 is dropped at random on the floor. What is the ...
2
votes
1answer
40 views

Geometric probability of intersection of a square and a circle

In the unitary square we choose a point $(X, Y)$ with iid coordinates $U [0,1]$ and a radius $R$, independent of $(X, Y)$ and $U [0,1]$, and we draw the circle of radius $R$ with center $(X, Y)$. Find ...
0
votes
1answer
16 views

Uniform Distribution: Probability that $X$ is rational

In Rosenthal's A First Look At Rigorous Probability Theory one of the phrases about a random variable $X$ having a Uniform Distribution from $0$ to $1$ is the following: ...But now suppose we ask, ...
0
votes
1answer
26 views

prove or disprove pivot quantity uniform distribution

let $X_i\sim U(0,\theta)$ indepedent uniform variables for $1\leq i \leq 5$ prove or disprove: $\frac{X_{1}+X_{2}}{\theta}$ is pivot quantity for $\theta$. I start with $P(\frac{X_{1}+X_{2}}{\theta}...
1
vote
0answers
14 views

Finding limits for uniform ratio distribution [duplicate]

Suppose that X∼ Uniform(a,b) and Y~ Uniform(c,d). Let Z =Y/X, I'm finding PDF Pr(Z$\leq$z). I took the region for integration as $ \frac{c}{a} \leq z \leq \frac{d}{a}$ ; $ \frac{d}{b} \leq z \leq \...
2
votes
2answers
70 views

Uniform Random Variable on $[0,1]$ and Bernoulli$(1/2)$

Let $X_1,X_2,...$ be independent, identically distributed (iid) random variables with distribution Bernoulli$(1/2)$. Define the random variable: $$Y=\sum_{n=1}^\infty\frac{X_n}{2^n}.$$ Then $Y$ is ...
2
votes
2answers
55 views

Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
1
vote
1answer
47 views

Solving the quadric equation: $\frac{-U\pm \sqrt{U^2-4V}}{2}$ where $U,V$~$\mathbf{U}(-1,1)$ independently.

I'm given two uniform random variables $V,U \sim\mathbf{U}(-1,1)$. I also get the function: $$h(s)=s^2+Us+V$$ I'm interested to answer queations like for which values $h(s)$ has only: zero /one/ two ...
0
votes
0answers
13 views

Convolution of 3 uniform random variables

I really do not know how to do this. Let X have a uniform distribution on (0,100) (time to failure from 0 hours to 100 hours), avg=50. I need to determine the distribution function of 3 components. ...
1
vote
1answer
43 views

Probability generating function of $X\sim \text{Poisson}(\lambda)$ when $\lambda\sim U(0,2)$

The probability generating function (pgf) of $X\sim \text{Poisson}(\lambda)$ is $$G_x(t) = e^{-\lambda(1-t)}.$$ Find pgf of $X$ if $\lambda\sim \text{Unif}(0,2).$ Then find $\mathbb P(X=2).$ My ...
0
votes
1answer
46 views

Distribution function of $\sin(\pi\theta)$ when $\theta\sim U(-1,1)$

If $\theta\sim Unif[-1,1]$, then what is the CDF of $U=\sin(\pi\theta)$? Now, its easy to see that $$P_{U}(t) = P\left(\theta \leq\frac{\sin^{-1}(t)}{\pi}\right)$$ somehow the answer is equal to : ...
2
votes
1answer
16 views

showing the pdf of n-th order statistics

I am working on a mathematical stats assignment and I got stuck here. Letting $X_1, X_2, ... ,X_n$ a random sample from uniform(0,$\theta$), and Y is n-th order statistic, I need to show that the ...
1
vote
1answer
63 views

Show that $((X_1,…,X_n)|X_1+\dots+X_n=t)$, $X_i \sim Exp(1)$ is uniformly distributed

Let $X_i \sim Exp(1)$ be independent. I need to show that $((X_1,...,X_n)|X_1+\dots+X_n=t)$ is uniformly distributed over all nonnegative vectors that sum to t. What does "over all nonnegative ...
0
votes
1answer
17 views

PDF of convoluted random variable conditional on another convoluted one

Suppose $V,W,$ and $X$ are mutually independent random variables. Further let $Y=V+W$ and $Z=V+X$. Is there a way to characterize the joint density $f_{Y,Z}(y,z)$ given the dependence of $Y$ and $Z$? ...
0
votes
1answer
26 views

I need to find the marginal distribution of Y from the following distributions

$f_X(x) = \frac{1}{2}e^{\frac{-x}{2}}$ and $f_{Y|X}(y|x) = I_{[0;x^2]}$ (Uniform continuous from $0$ to $x^2$). I tried finding the joint distribution by using $f(X,Y) = f(Y|X) * f(X)$ and then ...
0
votes
0answers
14 views

Generating random points in a square, such that any two points must be a certain minimum distance from each other

I want to generate random points in a square, such that any two points must be a certain minimum distance from each other. I know that one way to do this would be to simply generate uniformly ...
1
vote
0answers
31 views

p-vector uniform distribution in ball and $X_i$ i.i.d exponential distribution ($\lambda$) with $ \theta = E\{ X_1 -t ∣ X_1 \gt t \}$

There's some questions: First one: Suppose $X_1, \dots, X_n$ are p-vector uniform distribution in the ball $B_\theta = \{x ∣ \Vert x \Vert \lt \theta \} ;\theta>0 $ is an unknown parameter. ...
1
vote
1answer
49 views

Finding $P\left(X<\frac{3n}{2}\right)$ where $X$ is uniform on $\{n,n+1,\ldots,2n\}$ [closed]

If $X\sim \text{Uniform}\{n,n+1,\ldots,2n\}$, how can I find $P\left(X<\frac{3n}{2}\right)$ (in terms of $n$ where relevant) for both odd and even values of $n$? I got this in a test today and I ...
0
votes
1answer
54 views

Finding $h$ such that $h(U)$ has pdf $f(x) = \frac{3}{x^4} \mathbb{1}_{[1, +\infty]}$ when $U$ is uniform

Let $U$ be a random variable distributed uniformly in $[0,1]$. My task is to find function $h$ such that $Y = h(U)$ has density function denoted by $$f(x) = \frac{3}{x^4} \mathbb{1}_{[1, +\infty]}.$$ ...
3
votes
2answers
56 views

Expected Maximum of Three Numbers that Sum to 1

I've been working on a programming project on classification with 3 classes, and I'm interested in comparing my results to what I'd expect from pure noise. So I have the following question: Let $X,Y,...
2
votes
1answer
48 views

Prove that “equiprobable” is an equivalence relation

Let $X$ be a set of cardinality $n$ and let $R$ be the set of all relations over $X$. Consider the probability space $(R,P)$ with uniform distribution, that is, $P[\{R_1\}]] = P[\{R_2\}]$ for all $ ...
1
vote
1answer
13 views

Continuity corrections on modelling discrete distributions

A discrete random variable X has the distribution $U(11)$. The mean of $50$ observations of $X$ is denoted by $\bar{X}$ . Use an approximate method, which should be justified, to find $P(\bar{X} \...
0
votes
1answer
33 views

Constructing a specific iterative function

I have a problem at hand that i would like to share with giving my inputs and hoping for some from this site. so here it goes. I have a matrix say a $4\times 4$ matrix $$A= \begin{bmatrix} a_{11} &...
2
votes
1answer
29 views

Finding the expected value for various functions of two independent random variables

Let $X$ and $Y$ be independent random variables with uniform density functions on $[0, 1]$. Find: $E(|X-Y|)$ $E(X)=E(Y)=1/2$, $f_{X,Y}(x,y)=1$ Integrating the region where $x > y$ and using ...
0
votes
1answer
37 views

Which estimator would be better in terms of Mean Square Error?

Let $X_1,\ldots,X_n$ be a a sample from a Uniform Distribution $(0,\theta)$ where $\theta > 0$ is an unknown parameter. I have found the estimator based on the sample mean $$\hat{\theta}=2\bar{X}$...
0
votes
1answer
18 views

Adjusting a set of random numbers such that they approach a uniform distribution when biased noise is added [closed]

A good random number generator, $G$ will produce a sequence of $[0, 1]$ values which are near uniformly distributed as $n$ draws goes to infinity. If I start drawing samples from the random number ...
0
votes
3answers
57 views

Probability distribution difficult exercise [duplicate]

I have to solve this exercise and I have no idea how to do it... Help is highly appreciated. We make the following experiment: we ask 2 persons to write one real number from [0, 5] each on a ...
0
votes
1answer
36 views

Distribution of distance from origin for uniformly randomly chosen point in circle

So I think I know how to solve (a) correctly for this problem, but I keep getting answers to (b) that don't integrate to be $1$. I think (c) follows straightforwardly from there so (b) is the big ...
0
votes
0answers
24 views

How do I choose the boundaries when I am calculating the marginal $f_X(x)$ of a function?

So I have a question where I have a function: $$f_{X,Y}(x,y)=c$$ for $$(x,y) \in T$$ Where I have seen that $c$ = $\frac{1}{8}$ and where $T$ is the triangular region bordered by $x=0, y=0, x+y=4$....
0
votes
2answers
25 views

Uniform discrete distribution - time to draw

I have a question about the basic definition of discrete normal distribution. Let's assume I have a machine that draws a number ranging from 1 to 3 from a uniform discrete distribution (the ...
1
vote
1answer
53 views

$Y = \frac{X_1 X_2}{X_3}$ where $X_i$ is a uniform random variable

$Y = \frac{X_1 X_2}{X_3}$ where $X_i\sim U(0,1)$ and $X_1,X_2,X_3$ are i.i.d I need to calculate $Var(Y)$ and $Var[Y|X_3=1.7]$ I know that for each $X_i$, $E[X_i]=\frac{1}{2}$ $Var[X_i]=\frac{1}{...
1
vote
0answers
27 views

Central moment for a uniform distribution

The probability density function of T is given by $$f(t) = 1/2h \text{, for each } t\in(-h,h) $$ where $h > 0$. Derive an expression for the central moment I used integration and got $\frac{(b-u)...
1
vote
1answer
31 views

How to transform a $U(0,1)$ variable to produce a Poisson variable?

Suppose $ X $ is a uniformly distribution over $(0,1)$. How to find transformations $Y=g(X)$ to produce random variables with the Poisson distribution?
1
vote
1answer
32 views

Finding density function of random variable

Choose an uniformly distributed random variable $U$ on the unit interval $[0,1]$. Then, what is the probability density function of $Y= \ln(U+ 1)$? I know the density function is the derivative of ...
0
votes
1answer
26 views

How can calculate conditional pdf of Y when you dont know about f(y)

X is a uniform distribution on the interval (0,1). Y is a also uniform distribution on the interval (0,x). Its the only information that I could know. Then how can I calculate p(Y|x)? If you teach me, ...
0
votes
0answers
45 views

understanding the conditional entropy in the case of having uniform distribution?

Would you please help me to understand the conditional entropy in this example which I got stuck in? The example Considers 4 uniformly popular binary vectors, for example; {f1,f2,f3,f4} each with ...
0
votes
1answer
26 views

Uniform distributed success probability for a coin

$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
0
votes
4answers
55 views

Frog makes two jumps (uniform distribution)

I received this problem on my exam, although I thought I answered it right, it was marked as wrong. There is a frog on a line. The frog starts from a point 0 and makes two successive jumps: first ...
0
votes
0answers
30 views

P/1 Actuary Question: Expected value for continuous uniform distribution.

A question reads: A loss random variable has a continuous uniform distribution on the interval $(0, 100)$. A insurance policy on the loss pays the full amount of the loss if the loss is less than or ...
0
votes
2answers
36 views

Probability mass function of $ \min(X, Y)$ where $X,Y$ are i.i.d discrete uniform

Let $X$ and $Y$ be two discrete uniform i.i.d random variables distributed over $\{0, 1, 2,\ldots, N\}$. Find the pmf of $Z = \min(X, Y)$. From what I understand, I have to find the joint $pmf$ first,...
1
vote
1answer
68 views

Coverage probability for Uniform$(0, \theta)$

Let $X_1 \dots X_n$ denote a random sample from a uniform $(0, \theta$) distribution. PROBLEM: Compute the coverage probability for the CI: $$\left(\frac{X_{(n)}}{0.95}, \frac{X_{(n)}}{0.25}\right)...
1
vote
1answer
44 views

What is P($X_{1}>X_{2}>…X_{n-1}>X_{n}$)?

Given $X_{i}$, $1 \leq i\leq n$, are independent random variables with uniform distributions on $[0, 1]$, what is P($X_{1}>X_{2}>...X_{n-1}>X_{n}$)? I thought it would be something along the ...
0
votes
0answers
19 views

JPDF for uniform-distribution

you are given that the joint distribution of X and Y is uniform on the region defined by the conditions: $0<x<1$, $x<y<x+1$. Find the correlation coefficient of X and Y My problem with ...
1
vote
1answer
16 views

Alternate characterization of a spatial Poisson point process

Would I be correct to assess that a spatial Poisson point process on some compact, say the $d$-dimensional sphere, can be simulated by first choosing some $n \sim \mathrm{Poisson} (\beta)$ number of ...