All Questions
Tagged with density-function uniform-distribution
57 questions
9
votes
1
answer
365
views
Defining a uniform distribution over the perimeter of a polygon
Problem definition
Consider a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and let
\begin{equation*}
\begin{aligned}
z&=\underbrace{\sum_{j=1}^n \left[(V_{j+1}-V_j) \frac{L}{L_j}\left(\...
- 473
1
vote
1
answer
62
views
Expected Value for Complex-Valued Random Variable
This question is part of Exercise 3.14 in the book The Analysis of Time Series: An Introduction with R, 7th Ed., by Chatfield and Xing.
Problem Statement: Suppose $\theta$ is uniformly distributed ...
- 6,039
0
votes
0
answers
41
views
Uniform density over 2 segments [duplicate]
Background
Let $V_1, V_2 \in \mathbb{R}^2$ be the vertices of a segment and let $z$ be uniformly distributed over that segment. Now consider the random vector
\begin{equation*}
\begin{aligned}
y&=...
- 473
1
vote
0
answers
65
views
Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
- 345
3
votes
1
answer
281
views
Two dimensional random variable with uniform marginal probability density functions [duplicate]
I have access to some data for two variables - let's call them x and y.
In particular, I have the distribution of data separately per each variable, something that allows me to estimate the marginal ...
1
vote
1
answer
72
views
Uniform Sampling From the Region Bounded by $\sqrt{x}$, $x=3$, and $y=0$
I want to sample uniformly from the area bounded by $y=0$, $x=3$, and $y=\sqrt{x}$:
If I draw $x$ from $U[0, 3]$ and $y$ from $U[0, \sqrt{x}]$, the density will be higher in the bottom left corner:
...
- 1,099
0
votes
1
answer
235
views
Find the MLE density function of uniform [-\theta,\theta] [duplicate]
For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-...
1
vote
0
answers
73
views
Difference of dependent uniform random variables [closed]
My question is similar to the one posed here for a sum of dependent uniform RVs.
How can I find the CDF of $T=X-aY$, where $X\sim U[0,B]$, $Y\sim U[0,X]$, and $a<1$ is a constant. I've tried ...
- 11
4
votes
1
answer
147
views
How to see that this is a mixture of the uniform and Erlang?
I have a pdf in a form:
$$f(x) = \begin{cases}\frac{1}{2} + \frac{1}{2} \frac{\lambda^k x^{k-1} e^{-\lambda x}}{k!} & \text{for $0<x<1$} \\ \frac{1}{2} \frac{\lambda^k x^{k-1} e^{-\lambda x}}...
- 604
1
vote
0
answers
156
views
Convolution: PDF of difference of uniform random variables [closed]
PDF of $X$:
PDF of $Y$:
$Z=X-Y$, $T=X+2Y$, how to find the PDF of $Z$ and $T$ and plot them?
- 191
0
votes
1
answer
97
views
Simple notation question: pdf for mle of uniform?
I have simple notation question related to pdf for mle of uniform $U(0,\theta)$.
Given following pdf $f(\hat{\theta}_{MLE}) = \frac{n \cdot \hat{\theta}_{MLE}^{(n-1)}}{\theta^n} $ , I'm confused ...
- 251
0
votes
1
answer
639
views
Which has minimum concentration: the uniform distribution or the maximum entropy distribution?
For a continuous random variable, the uniform distribution has high entropy because it demonstrates the greatest level uncertainty.
However, this conflicts with the maximum entropy principle, which ...
- 4,049
2
votes
1
answer
209
views
Pdf of the sum of two independent Uniform R.V., but not identical
Question. Suppose $X \sim U([1,3])$ and $Y \sim U([1,2] \cup [4,5])$ are two independent random variables (but obviously not identically distributed). Find the pdf of $X + Y$.
So far. I'm familiar ...
- 384
2
votes
1
answer
70
views
Domain problem when calculating marginal density
I have the following homework assignment: the life expectancy $X$ of a lamp has exponential distribution with rate $\lambda$. The rate depends on the production proccess, such that its population can ...
2
votes
1
answer
382
views
What's the expression for convolution of a uniform[a,b] density and a normal(0,d^2) density?
Suppose I have $X\sim Uniform[a,b]$ and $Y\sim normal(0,d^2)$, what's the expression for the density of $Z=X+Y$?
Let $F_{Z}(z)$ be the cdf of $Z$ evaluated at $z$, and let $\Phi(\cdot)$ and $\phi$ be ...
- 3,050
4
votes
1
answer
2k
views
Does minimizing KL-divergence result in maximum entropy principle?
The Kullback-Leibler divergence (or relative entropy) is a measure of how a probability distribution differs from another reference probability distribution. I want to know what connection it has to ...
- 4,049
1
vote
1
answer
161
views
finding PDF of Y, given Y|X [closed]
$$Y|X\sim Bin(X,n)$$
$$X\sim U([0,1])$$
How can I find the PDF of Y?
I know that:
$$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
- 43
1
vote
1
answer
1k
views
Density of square root of sum of squared independent uniform random variables [duplicate]
Let $X \sim U(-1, 1)$ and $X \sim U(-1,1)$. We want to find density function of $W = \sqrt{X^2 + Y^2}$.
I got stuck and I have no idea, where I am making a mistake. This is my approach.
Let $F$ be a ...
- 325
1
vote
1
answer
803
views
Why does Uniform distribution make sense?
This might be a dumb question, but I am suddenly confused on how to understand the PDF of a uniform distribution.
For instance, the PDF of standard uniform is always equal to 1... How is that ...
- 13
5
votes
2
answers
2k
views
If $X=\sin\Theta$ and $Y=\cos\Theta$ with $\Theta$ uniformly distributed, how can I compute the joint pdf of $(X,Y)$?
I have a random variable $\Theta$ uniformly distributed between $[-\pi ,\pi]$, two functions $X=\sin\Theta$ and $Y=\cos\Theta$. I know that $X$ and $Y$ are uncorrelated but not independent. I want to ...
- 83
1
vote
1
answer
425
views
PDF of cosine of a uniform random variable with additional shift
I need to calculate the PDF of a random variable, which is quite similar to what was asked here. However, I have to deal with a shifted cosine function. Thus, my random variable is defined as
$$Y:=cos(...
- 113
3
votes
1
answer
1k
views
Sum of exponential of uniform random variables?
Let $F_{i}$ and $\phi_{i}$ are uniformly distributed independent random variables in the range $[-50,50]$ and $[-\pi/4,\pi/4]$, respectively.
If $N = 10$ and
$$Z = \sum_{i=0}^N e^{j(F_{i}+\phi_{i})}...
- 121
2
votes
1
answer
579
views
How to estimate the PDF of the logarithm of a uniformly distributed random variable?
This is a question I have to solve and need help with. I know it's usual to give pointers and hints so the OP can follow from there. Thus, I'll appreciate all input that shows me the way to go.
Let $...
- 706
3
votes
1
answer
9k
views
Sufficient statistics in the uniform distribution case
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
- 2,204
1
vote
1
answer
296
views
Sum of two continuous random variables
Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2).
What is the probability of R1>1 given that R1 +R2<2?
--
What I've tried:
I know that
$$
P(R1&...
- 439
1
vote
1
answer
868
views
Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function
I am presented with the following notes on Bivariate distribtions:
If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a marginal ...
- 2,204
3
votes
1
answer
3k
views
Calculating the sum of dependent uniform random variables
My question derives from Problem calculating joint and marginal distribution of two uniform distributions.
So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
- 153
-1
votes
1
answer
800
views
The pdf of a standard uniform random variable divided by constant [closed]
For a random variable $\frac{U}{a}$ where $U$ is a standard uniform random variable, I'm trying to determine the pdf. I'm not so sure what I'm getting is correct as I'm getting some funny results ...
- 321
4
votes
1
answer
1k
views
Transform X to get Y such that Y has a Uniform(0,1) distribution
A random variable $X$ has the PDF
$f_X(x) = \frac{x - 1}{2}, \ 1 < x < 3$
Find a monotone function $u(x)$ such that the variable $Y = u(X)$ has the distribution $Uniform(0,1)$.
- 535
2
votes
2
answers
411
views
What is the ratio of a N[0,1] and U[-1/2,1/2]?
I have come across a problem where I can reasonably assume that the numerator is a uniform distribution of the type U[-a,a], i.e., centered on zero, and the denominator is N[0,b]. This seems to be ...
- 13.3k
-2
votes
2
answers
1k
views
Uniform Density Function
As we know the uniform probability density function is
f(x)=1/(b-a)
if i find the density function and area of this uniform distribution between
(0, 1/2) then it would be
f(x)=1/(1/2-0)
f(x)=2
...
- 85
-1
votes
1
answer
4k
views
PDF of the maximum likelihood estimator of a uniform distribution
Suppose $ \{X_1, \dots , X_n \}$ is a random sample from:
$$
f_X(x; \theta) = \frac{1}{\theta} \text{, for } 0 \leq x \leq \theta
$$
The Likelihood function is easy to calculate:
$$ L_Y(\theta; y)...
- 107
2
votes
1
answer
155
views
Transforming a uniform PDF to a Gaussian PDF
I have a Uniform PDF from [-50, 50], I would like to transform it to a Gaussian. The methods that I read up about doing this(like Box Mueller) assume that the uniform distribution is between [0,1). Is ...
9
votes
1
answer
14k
views
PDF of cosine of a uniform random variable
There is a formula for the density of the cosine of random variable that's a uniform on $(-\pi,\pi)$ as discussed in this page: $f_{Y}(y) = \dfrac{1}{\pi \sin(\cos^{-1}y)}, y \in\ [-1,1]$
Can anyone ...
- 93
0
votes
1
answer
3k
views
Mean and Variance of the Area of a Circle with Uniform Radius
A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.(a) Find the mean and variance of A, without first finding the CDF or PDF of A.(b) Find the CDF and PDF of A.
So, quite ...
3
votes
1
answer
100
views
Showing Independence for X and frac(X + Y)
Suppose that we have independent samples $X_1, X_2 \sim \text{unif}[0, 1)^d$. I'm asked to show that $Y_1 = X_1$ and $Y_2 = X_1 + X_2 - \lfloor X_1 + X_2 \rfloor$ are also independent uniform samples ...
- 133
8
votes
2
answers
1k
views
How is the spherical elevation angle distributed when $(x,y,z)$ are uniformly and normally chosen?
As a follow up to
How the polar coordinate, $\theta$, is distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ and if $(x,y) \sim N(0,1)\times N(0,1)$?
Assume $(x,y,z) \sim U(-10,10) \times U(-10,10) \...
- 739
21
votes
3
answers
8k
views
How is $\theta$, the polar coordinate, distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ vs. when $(x,y) \sim N(0,1)\times N(0,1)$?
Let the Cartesian $x,y$ coordinates of a random point be selected s.t. $(x,y) \sim U(-10,10) \times U(-10,10)$.
Thus, the radius, $\rho = \sqrt{x^2 + y^2}$, isn't uniformly distributed as implied by $...
- 739
0
votes
0
answers
231
views
Characteristic function of uniform random variable [duplicate]
I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the ...
- 51
3
votes
0
answers
122
views
Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?
I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by
$$ \pi_G(\xi|\mu,\sigma) = {1\...
- 724
1
vote
1
answer
69
views
Discrete Distribution
In the die-coin experiment, a fair, standard die is rolled and then a fair coin is tossed the number of times showing on the die. Let N denote the die score and Y the number of heads.
a)I want to ...
- 1,226
3
votes
1
answer
2k
views
Multi-variate uniform distribution
Suppose that $(X,Y,Z)$ is uniformly distributed on $\{ (x,y,z) : 0 \leq x \leq y \leq z \leq 1 \}$
a. I want to find out joint density function of $(X,Y,Z)$.
b. I want to find out probability ...
- 1,226
2
votes
1
answer
354
views
How to compute the CDF of this random variable?
I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...
- 99
4
votes
2
answers
373
views
What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables
$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs.
...
- 347
49
votes
5
answers
42k
views
Why is the CDF of a sample uniformly distributed
I read here that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution.
I have ...
- 493
2
votes
1
answer
721
views
Probabilities of conditional expectation values in uniform distribution
Let's consider a continuous random variable $X$ as follows:
$f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\
\frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$
...
- 271
2
votes
1
answer
118
views
Why small values produce undulating densities when ploting logarithm of a loguniform prior (in R)?
I am using a program that draws random values in a log-uniform distribution let say between 1 and 100.
When I plot the density of the produced values with R it looks like a log-uniform distribution ...
- 233
8
votes
2
answers
811
views
PDF of a sum of dependent variables
This is a direct continuation of my recent question. The thing that I actually want to get is the distribution of $a+d+\sqrt{(a-d)^2+4bc}$, where $a,b,c,d$ are uniform in $[0,1]$. Now, the ...
- 1,264
17
votes
2
answers
689
views
What's the distribution of $(a-d)^2+4bc$, where $a,b,c,d$ are uniform distributions?
I have four independent uniformly distributed variables $a,b,c,d$, each in
$[0,1]$. I want to calculate the distribution of $(a-d)^2+4bc$. I computed the distribution of $u_2=4bc$ to be $$f_2(u_2)=-\...
- 1,264
3
votes
1
answer
4k
views
Show that $\min(U,1-U)$ and that $\max(U,1-U)$ are uniform
Let $U$ be uniform on $(0,\ 1)$. Show that $\min(U,\ 1-U)$ is uniform on $(0,\ 1/2)$ and that $\max(U,\ 1-U)$ is uniform on $(1/2,\ 1)$.
I'm not sure how to approach... the only hint i have is that a ...
- 41