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.
1,447
questions
0
votes
1answer
21 views
Find the a posteriori probability? (Ch-4,Exercise-21, Probability, Random Variables and Stochastic Processes-Papoulis)
The probability of heads of a random coin is a random variable p uniform in the interval (0,
1).
(a) Find P{O.3 <= P <= O.7}.
(b) The coin is tossed 10 times and heads shows 6 times.
Find the a ...
0
votes
1answer
44 views
Is $3-4U$ equivalent to $4(1-U)$?
If $U$ is defined to be the uniform distribution on $(0,1)$, is it true that
$3-4U\sim4(1-U)$?
0
votes
1answer
19 views
Is this an equivalent uniform distribution?
Suppose $U$ is uniformly distributed on $(0,1)$ then it is true that $X=\ln(1-U)\equiv \ln(U)$. This makes sense because $1-U\sim U$, and thus $X$ will have the same distribution. Now, suppose $c\in(0,...
1
vote
1answer
19 views
Variance of (X + Y)^2 where both X and Y are uniformly distributed between 0 and 1?
I can't seem to figure out how to solve that. If X and Y are two independent random variables X and Y sampled
uniformly from the [0, 1], then what is the variance of $(X+Y)^2$ ?
I know that $Var((X+Y)...
5
votes
2answers
87 views
Expected number of regions with $n$ random lines in a circle
There are $n$ random lines drawn in a circle, defined by endpoints being uniform on circle. I am trying to figure out the expected number of regions separated by $n$ lines. I know $f(0)=1$, $f(1)=2$ ...
0
votes
2answers
47 views
Distribution of $Y=X(1-X)$ when $X\sim U(0,L)$
I was wondering if someone could provide a hint of how to solve the following problem. I am working on some of my own research and can't seem to remember how to solve a problem like this:
Let $X\...
1
vote
1answer
24 views
conditional distribution of sum of order statistic
When I read Ross's book "Statistic Process" , I find the lemma, but I cannot prove it.
The lemma states that Let $Y_1, \cdots, Y_n$ be iid nonnegative random variables then $E[Y_1+\cdots + Y_k| Y_1+\...
2
votes
1answer
62 views
The density function of the sum of n independent uniformly distributed random variable on $(-1,1)$ is supported on $[-n,n]$
This question arose from the statement after Exercise 3.3.6 of Durrett's probability.
Exercise 3.3.6: Show that if $X_{1}, \cdots, X_{n}$ are independent uniformly distributed on $(-1,1)$, then for ...
0
votes
0answers
54 views
Universality of the uniform in the context of the Rayleigh distribution
I am currently exploring a theorem called universality of the uniform:
Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that ...
0
votes
1answer
24 views
Let $X$ be an r.v. with CDF $F$. Then $F(X) \sim \text{Unif}(0,1)$?
I recently encountered a theorem called universality of the uniform:
Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that ...
0
votes
1answer
17 views
Uniform distributions: location-scale transformation
My textbook, Introduction to Probability, first edition, by Blitzstein and Hwang, says the following:
In a location-scale transformation, starting with $X \sim \text{Unif}(a, b)$ and transforming ...
0
votes
1answer
42 views
Showing independence of ratio of ordered statistics for uniform distribution
From An Intermediate Course in Probability by Allan Gut
Suppose that $X\in U(0,1)$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be the order variables corresponding to a sample of $n$ independent ...
1
vote
1answer
61 views
Let $U_1,U_2,…$ be a sequence of independent uniform $(0, 1) $random variables, question about $\Pr(N > n)$
Let $U_1,U_2,...$ be a sequence of independent uniform $(0, 1) $random variables and let
$$N:=\min\{n\geq 2: U_n>U_{n-1}\}$$
$$M:=\min\{n\geq 2: U_{1}+\cdots+U_n>1\}$$
Show that ...
1
vote
1answer
29 views
What is the distribution when you add exponential rvs to uniform rvs and count?
Let $X_1, \dots, X_c$ be independent uniform random variables on $[0,1]$. We know the number of random variables whose value will fall in the range $[0,1]$ will always be exactly $c$. Now define $...
0
votes
2answers
67 views
Generating Uniformly Distributed Random Points in a circular region of a hyperbolic plane using this coordinate system
Let's say that in a hyperbolic plane we use a coordinate system, in which we have a u axis and a v axis that are both mutually perpendicular to each other. The coordinate lines that define u ...
0
votes
1answer
81 views
Finding pdf of $\tan(X)$ when $X \sim U (-\pi ,\pi)$
This question is a particular case of the following:
Suppose that X ∼ U ( $− π/2$ , $π/2$ ) . Find the pdf of Y = tan(X).
I came up at the same solution in the following post, ignoring for a moment ...
1
vote
0answers
33 views
Average of a set of probability measures
Given an integer $n\ge 1$, let
$
\textstyle \Delta:=\left\{x \in [0,1]^n: \sum_{i\le n}x_i=1\right\}
$
be the $n-1$-dimensional simplex. Consider a random variable
$
X: \Omega\to \Delta
$
(on a ...
0
votes
0answers
28 views
Find the joint distribution of the original variables with the largest two order statistics
$k$ independent variables $X_1, X_2,...,X_k$, each has a distribtion with CDF $F_1(x), F_2(x),...,F_k(x)$. Each time we randomly take a sample of each variable (denoted as $x_1, x_2,...,x_k$) and ...
3
votes
0answers
22 views
Expected value of general uniform order statistics
I know that if $X_k$ ~ $Unif(0,1)$ and is order statistics, then $E[X_k] = \frac{k}{n+1}$.
What's $E[X_k]$ for when $X_k$ ~ $Unif(a,b)$?
I think it's $a + \frac{k}{n+1}(b-a).$ Can someone confirm?
1
vote
0answers
62 views
Product of two discrete uniform distributions
Let $a$ and $b$ be positive integers with $b > a$.
Let $A$ and $B$ be two discrete uniform distributions over the integer intervals $[[1, a]]$ and $[[1, b]]$ respectively.
Then, given any $t \in [...
2
votes
1answer
130 views
Twist on classic interview question. $P(X > 3Y)$ where $X,Y$ are uniform random variables
So there's this very classic probability question that says:
Given$ X, Y$ two INDEPENDENT uniform random variables in $[-1,1]$, what is $P(X > 3Y)$?
Of course, there are alterations where the ...
0
votes
3answers
21 views
How do you compute uniform distribution with only mean and proportion given?
Machine A produces mints that have a label weight of 50g and is believed the weights of the weight is uniformly distributed, with a mean of 51.5g and 70% of them less than 52.5g.
What's the ...
1
vote
0answers
13 views
Sort a set of arrays such that the mean-member delta of each array position is minimized
Given n number of arrays A in the format An = [xn, yn, zn] where X, Y, and Z are collections in the format X = [x1, x2, x3, ...xn], Y = [y1, y2, y3, ... yn], and Z = [z1, z2, z3, ...zn], create a ...
0
votes
0answers
32 views
Unbiased estimator for UNIF($-\theta, \theta$)
I am searching for an unbiased estimator for $\theta$ in a UNIF($-\theta, \theta$) distribution, which looks like $\hat\theta = c(X_{n:n} - X_{1:n}$). The question is to search for the c that makes ...
2
votes
1answer
67 views
Density of $ Y = X + \frac{1}{X}$ when $X\sim U(a,b)$
Let $ X $ be a continuous random variable uniformly distributed in $ \left[a, b\right] $, that is $X \sim U(a, b)$. Suppose $ 0 < a < b $.
We wish to find the density function of $$ Y = X + \...
0
votes
0answers
21 views
Joint distribution of ratio of uniform distribution
If $X \sim \mathrm{Uniform}[X_{\min}, X_{\max}]$ and $Y \sim \mathrm{Uniform}[Y_{\min}, Y_{\max}]$, then what does the p.d.f of $Z = X^a/Y$ look like? I think it is a function that is segmented.
...
1
vote
1answer
35 views
Is there a notation for the distribution sampled from a set
Suppose we choose uniformly from the set S, is there a well accepted notation for this distribution? I imagine something like $X \sim U(S)$ or $X \sim Uniform(S)$? I want to use this to write that I ...
0
votes
0answers
19 views
Uniform sampling under constraints
I have a problem which is a bit weird. At some point in my program I have to sample some kind of mapping under constraints, uniformly at random.
Let's say I have a certain number of unique objects, $...
0
votes
1answer
48 views
Compositions of a large integer
Let's say $n$ is the number of integers and $t$ is their sum and we want to find all the combinations of number $t$, such that there are $n$ components. See https://en.wikipedia.org/wiki/Composition_(...
1
vote
0answers
44 views
Expected remaining time for bus to come, given that already waited for 5 minutes?
There are two parts, one for when time is uniformly distributed and then an exponential distribution.
Time is uniformly distributed from 0 to 10. If already waited
5 minutes, what is the expected ...
2
votes
1answer
26 views
$n^2$ times an equidistributed sequence
Suppose you have an equidistributed sequence $\{\alpha_n\}_{n=1}^\infty\subseteq(0,1)$, does $n^2\alpha_n\to\infty$ always? I don't know how to even start. Is it something like "it diverges for almost ...
0
votes
1answer
38 views
Density of Uniform distribution with respect to standard-normal distribution
How do I calculate the density function of the uniform distribution $U_{a,b}$ with respect to standard-normal distribution $N(0,1)$?
0
votes
1answer
15 views
Metric for uniformness of distribution of points in an irregular shape
I am looking for a mathematical way to check if the distribution of points inside some region (almost never a proper form) are evenly and uniformly distributed through it. Do you think this is ...
3
votes
1answer
113 views
A stick is broken and its left part is discarded.Probability that one of them $>1$ [duplicate]
A stick of length $2$ m is made of uniformly dense material. A point is chosen randomly on the stick and the stick is broken at that point. The left portion of the stick is discarded and now again ...
0
votes
1answer
26 views
Probability of $P\left(X \in \left[\frac{a + 3b}{4}, b \right] \right)$ for uniform distribution
For $X ∼ U(a,b)$, with $a,b > 0$, what is $P\left(X \in \left[\frac{a + 3b}{4}, b \right] \right)$?
I do not know how to solve this.
Does $$P(X \in [(a + 3b)/4, b]) = P((a + 3b)/4 < X < b)~?...
0
votes
1answer
48 views
Figuring Out Marginal Density by Looking at Plot of Joint Density
I have the below plot of the joint density of X and Y.
X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on values between 0 and 1.
Can someone please explain ...
3
votes
1answer
155 views
Euclidean distance of two uniform random variables
Two random variables $X$ and $Y$ are uniformly distributed, the pdfs of which are given by $f_{X}\left(x\right) = f_{Y}\left(y\right) = 1/r$. I am trying to obtain $Z = \sqrt{X^2 + Y^2}$.
I tried the ...
0
votes
2answers
35 views
Joint probability density for non-identical Uniform random variables
Let $~X,~ Y~$ be uniform on $~[0, 3] × [2, 4]~$. Find $~P(X + Y ≤ 5)~$ and
$~X~$ and $~Y~$ are independent.
My approach: Using convolution formula.
Difficulty I am facing: Understanding the limits ...
0
votes
2answers
52 views
What Distribution Summed is Uniform?
The sum of n independent uniform random variables forms a new random variable having an Irwin–Hall distribution with parameter n. (Approximately normal when n is large)
$$X_i \sim \cal{U},\quad \...
1
vote
0answers
43 views
Does there exist a uniform distribution on the set of all permutations of a countably infinite set?
For a finite set $N = \{1,2,3, \ldots, n\}$, a permutation over $N$ is a bijective function $\pi: N \to N$. A uniform distribution over the set of all permutations of $N$ must assign each permutation $...
-7
votes
1answer
35 views
How to predict most possible value from a set, where values don't repeat? [closed]
I have a set of readings.
For example, R={69,70,73,65,170}
How can I find which might be more accurate value? (It can be out of the set)
3
votes
1answer
74 views
Conditional expectation of sum of two uniform random variables
I want to compute the conditional expectation of the sum of two independent uniform random variables, with two conditions. Is there anything wrong with my approach below?
Formally, let $X,Y$ both be ...
2
votes
1answer
28 views
Construct an open set containing all rational points in interval with given measure.
In a recent exam I was asked to construct an open set $M \subseteq [0,1]$ mit $\mathbb{Q} \cap [0,1] \subseteq M$ with $\lambda[M] \in (0, \frac{1}{2})$ where $\lambda$ is the uniform distribution on $...
1
vote
2answers
67 views
MLE for uniform distribution around $[-\theta,\theta]$ [duplicate]
Given $X_1,\ldots,X_n$, where $X_i\sim U(-\theta,\theta)$, what the MLE for $\theta$?
Apparently the answer is $\max\{|X_1|,\dots,|X_n|\}$ but I can't figure out why.
The density function is
$$f(x,\...
0
votes
1answer
30 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 $\zeta(s):$ $$\mathbb{P}(X_s=k)=\...
0
votes
0answers
46 views
Minimal Sufficient Statistic for $U(0, \theta)$
The definition of a Minimal Sufficient Statistic (MSS) denoted $S(X)$ is
$$
\frac{L(\theta;x)}{L(\theta;x)} \text{ independent of $\theta$} \iff S(X) = S(Y),
$$
assuming the densities exist and $L$ ...
0
votes
2answers
36 views
Finding CDF and PDF of $Y=20/X$ when $X$ is uniform on $[4,7]$
I have a problem where $X$ is uniform on the interval $[4,7]$ and $Y = 20/X$.
I am asked to find $F_Y(y)$ and $f_Y(y)$ using the CDF and PDF.
This is a uniform distribution, so it's easy enough ...
6
votes
1answer
150 views
Uniformly Distributed Random-Variable With Specific Ordering
Let $0\leq a<b$. Define the subset of $[a,b]^n$ by
$$
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
31 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 ...
