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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.

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Derivation of the expected minimum distance between uniform random variables on the unit interval

Suppose we have $n$ iid random variables, $U_1,...,U_n\sim\text{Unif}(0,1)$. What is the expected minimum distance between any two of these random points? Now, I know this has already been asked and ...
aaaaaaaaaaaaaaaaaanon's user avatar
1 vote
2 answers
121 views

Optimal strategy for uniform distribution probability game

There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
jimsimons's user avatar
2 votes
2 answers
103 views

The distribution of $XY+(1-X)(1-Y)$ for $X,Y$ sampled uniformly from [0,1]

Let $X,Y$ be sampled uniformly from the interval $[0,1]$ and $Z=XY+(1-X)(1-Y)$. I would like to know the exact distribution of $Z$. I conjecture it should be uniform as well, but was not able to prove ...
user50394's user avatar
  • 429
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0 answers
25 views

Hash Table and uniformly distribution

I'm trying to understand the relationship between input distributions and hash function performance in hash tables. In statistics, the theoretical probability of landing in a specific slot of a hash ...
miiky123's user avatar
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0 answers
34 views

Probability of Christmas falling on a particular day of the week

What is the probability that Christmas (on a randomly chosen year) falls on a Monday, Tuesday, Wednesday, &c.? Accounting for the leap years and the 400-year repeating cycle, I wrote a program to ...
Nína's user avatar
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0 answers
44 views

Uniform random variables with distribution over [1, 4]

2.4.2 Let W~Uniform[1 4]. Compute each of the following. (b) P(W >= 2) (c) P(W^2 <= 9) For b, I was thinking P(2<=w<=4)= (4-2)/(4-1) = 2/3. For c, I was thinking since w^2 <= 9, P(1<=...
hsigmon's user avatar
2 votes
2 answers
248 views

Uniform on $(0,1)$ vs Uniform on $[0,1]$

I read many papers about probability integer transformation or copula. Regarding the probability integer transformation, some use $X \sim U(0,1)$, and others use $X \sim U[0,1]$. What are the ...
Dr. Statistics's user avatar
0 votes
1 answer
55 views

If $X$ is a uniformly distributed discrete random variable, what is the condition that $Y=\phi (X)$ is too?

More specifically I was solving the following problem: Let 𝑋 be a discrete random variable that is uniformly distributed over the set $S=\{−10, −9, ⋯ , 0, ⋯ , 9, 10\}$. Which of the following random ...
Awe Kumar Jha's user avatar
0 votes
1 answer
28 views

Finding Expectation of a Uniform random variable from its moment generating function

From Taylor series, if I need to get the k-th moment, I need to find the k-th derivative of the Moment Generating function. If I have $X \sim \text{Uniform}(0,1)$, the MGF $M_{X}(s)$ is; $$M_{X}(s) = \...
moseskabungo's user avatar
1 vote
1 answer
68 views

Distribution of random variable when lower/upper limit is known

Let $X$ be an i.i.d. random discrete variable in $[0,255]$. That is, $\mathrm{P}(X=x)=\frac{1}{256}$ and the distribution looks something like What is $\mathrm{P}(X=x\mid X > 127)$? I imagine the ...
Cutaraca's user avatar
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1 answer
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What does this author mean by a simple compactness argument?

In the book "Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis", in the first chapter they say that a sequence of points {$u_n$} is uniformly distributed if: ...
FoxToast's user avatar
1 vote
2 answers
74 views

Conditional Expectation of $X/Y$ given $X \leq 2Y$ for Independent Uniform Variables

Let $X$ and $Y$ be independent random variables, each uniformly distributed on $(0,1)$. Find the conditional expectation $\mathbb{E}\left[\frac{X}{Y} \mid X \leq 2Y\right]$. I've tried to solve this ...
Fernand's user avatar
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0 answers
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The distribution of X + Y - floor(X + Y) where X and Y are independent and uniformly distributed over (0, 1)

I'm having trouble understanding the following discussion on the distribution of $X + Y - \lfloor X + Y\rfloor$ where $X$ and $Y$ are independent and uniformly distributed over $(0, 1)$: If $t \in [0, ...
johnsmith's user avatar
  • 367
1 vote
0 answers
50 views

How to determine if a race course is fair (on the open water)?

I am analyzing the distribution of race results using statistical methods like chi-square tests to look for the most uniformly distributed results. I have data from a number of different locations and ...
user6972's user avatar
  • 153
2 votes
2 answers
44 views

Probability that maximum of two iid Unif(0, 1) r.v.s is less than the minimum of two other iid Unif(0,1) r.v.s?

Specifically, let $X_1, X_2, X_3, X_4$ ~ $Unif(0, 1)$. What is $$P(max(X_1, X_2) \lt min(X_3, X_4))$$ Similarly, what is $$P(min(X_1, X_2) \gt max(X_3, X_4))$$ ?
gloves-'s user avatar
  • 21
1 vote
0 answers
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How can we fill the integration bounds and define a function $s$ from the given data so that this double integral is the mean of this indefinite int?

Suppose $f:\mathbb R \to \mathbb R$ has $\int_{-\infty}^\infty f(t)\ dt = 1 \in \mathbb R$. Define $g : \mathbb R \to \mathbb R$ by $g(\alpha) = \int_{-\infty}^\alpha f(t)\ dt$ and define $h : \mathbb ...
Snared's user avatar
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1 vote
0 answers
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Gaussian line integral over a polygon

Problem definition Let $y=z+v$, where $z$ is uniformly distributed over the contour of a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and $v\sim\mathcal{N}(0,R)$. Let $\ell$ be the set of ...
matteogost's user avatar
1 vote
1 answer
69 views

Normalized hamming distance in probability

Let two functions $f,g :[n]\to \{0,1\},$ then we define $$\delta{(f,g)}=\frac{|\{i\in[n]:f(i)\neq g(i) \}|}{n}$$ is called normalized hamming distance. My teacher said me this $\delta{(f,g)}$ is ...
user avatar
0 votes
1 answer
39 views

Probability of q location of n length string

Suppose we have an algorithm A that looks at only $q$ locations in the binary string on length $n$. Let we have two strings which is differ by only one bit. If we give these two strings to A , it will ...
user avatar
1 vote
1 answer
54 views

Finding $\mathbb{E}[\max(|\xi - \eta|,|\xi - ζ|,|\eta - ζ|)]$

$\xi, \eta, ζ$ - independent random variables uniformly distributed on the interval [0, 1]. Using conditional expected value, find $\mathbb{E}[\max(|\xi - \eta|,|\xi - ζ|,|\eta - ζ|)]$ Here what I got:...
Arthur_Kitsuragi's user avatar
1 vote
1 answer
61 views

Chernoff bound of strings

Let $x\in(0,1)^*$ be a any binary string. Consider $m$ sample indices $i_1,i_2,\dots,i_m$ uniformly at random and independently of one another. Each $i_j$ is sampled uniformly at random from $[|x|]$ ...
user avatar
2 votes
1 answer
65 views

$\mathbb E(\max(X_1,...,X_{t+1})|\mathcal{F}_t)$ where the $X_i$ are iid uniform

Let $X_1,...,X_T$ be independent and identically distributed uniform random variables on $[0,1]$. Let $$M_t:=\max\{X_1,...,X_t\},$$ $L_t=M_t-ct$ for a $c>0$ and $L_0:=-\infty$. If $\mathcal{F}_t=\...
Analysis's user avatar
  • 2,482
0 votes
0 answers
47 views

Probability that the Largest of Three Independent Uniformly Distributed Random Variables Exceeds the Sum of the Other Two

I am currently studying probability and statistics and I came across a problem that I am having trouble with. I have some understanding of random variables and their distributions, but this particular ...
prob1 yuma's user avatar
2 votes
0 answers
15 views

Expectation of a piecewise const approximation based on Beta distribution

Let $X_1, X_2 \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$ and then sort $X_1,X_2$ to get $X_{(1)} < X_{(2)}$. Based on the pdfs of $X_{(i)}$, we know $X_{(1)} \sim \mathrm{Beta}(1,2)$ and $...
learner's user avatar
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0 votes
1 answer
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Probability that a uniformly distributed point lies within a triangle in the unit disc

Question: Consider $\|{z}\|\le1$ in the complex plane (ie a disc radius 1 centred at the origin). The points X and Y are independently & uniformly distributed within the disc. A third point W is ...
edster101's user avatar
1 vote
1 answer
44 views

Resampling techniques to obtain uniform random variables.

Suppose we have $n$ iid samples $X_i$ from a distribution, say a triangular distribution symmetric about the origin, with support $[-h,h]$, $h>0$ (denote this density $f(x)$). I would like to ...
xyz's user avatar
  • 1,022
0 votes
1 answer
36 views

Discrete Convexity of a Function

Let $Y \mid p \sim \text{Binomial}(x, p)$ and $Z \sim \text{Uniform}\{a, b\}$, where $a$, $b$, $x$ are non-negative integers and $a < b$. I am trying to prove that for any fixed probability $p$, $\...
Md Hishamur Rahman's user avatar
0 votes
0 answers
19 views

Finding conditional expectation when variables are drawn from uniform distribution with different edges

I have the following problem - $a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $(m, 1+m)$ and $b$ is drawn from uniform distribution $(0,1) $. ...
Elina Gilbert's user avatar
1 vote
1 answer
29 views

Exponential Moment of Uniform Random Vector

Let $Z = (z_1,\dots, z_d)$ with $z_i$ i.i.d. uniform random variables on the interval $[a,b]$. I am interested in computing $$ \mathbb{E} [\exp (\|Z\|^2)] = \mathbb{E} [\exp (z_1^2 + \dots z_d^2)]. $$ ...
WeakLearner's user avatar
  • 6,106
0 votes
1 answer
20 views

Trouble proving inequality regarding the modulus of a complex number sum in uniform distribution theorem

I'm currently working through the proof of the theorem stated in Uniform Distribution of Sequences by Harald Niederreiter and L. Kuipers. The theorem states that if $\Delta x_n = x_{n+1} - x_n \...
dapsone_parrot's user avatar
0 votes
0 answers
52 views

Help with understanding the calculation of an expected sum

A box with $N$ balls numbered from $1$ to $N$. We take $n$ balls out with no returning. I need help with understanding the calculation of their expected sum. So let's say that $S_n$ is the sum of the ...
Jonathan Rosh's user avatar
2 votes
0 answers
93 views

Can we sequentially rotate a die on 3 axes from a given starting position so that the result is uniform?

Edit It seems that my initial assumption that the rotations in question occur simultaneously was wrong. They are calculated sequentially but animated simultaneously. Therefore I have updated the ...
pawello2222's user avatar
9 votes
2 answers
338 views

Probability that Mercury is the nearest planet to Earth.

Motivation: We tend to think of Venus as the nearest planet to Earth because at its nearest approach to Earth, Venus is the closest at 39 million Km away. This is followed by Mars at 56 million Km and ...
Nilotpal Sinha's user avatar
0 votes
1 answer
71 views

Conditional Expectation when variables are drawn from uniform distribution of different domain

I have the following problem - $a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $\{m, 1+m\}$ and $b$ is drawn from uniform distribution $\{0,1\}$. ...
Elina Gilbert's user avatar
6 votes
0 answers
240 views

Expected value of fastest career hat-trick?

This is a sports variation of line segment (rope, stick...) divided into $n+1$ subsegments by $n$ uniformly distributed points. Assuming goals are uniformly distributed among player's career, what's ...
Ilyan's user avatar
  • 61
1 vote
1 answer
51 views

Uniform distribution problem solving

On a street segment p of length $13 m$, a car of length $5 m$ parks completely at random. What is the probability that another car of the same length can fit? I am not sure whether I am correct but ...
Need_MathHelp's user avatar
3 votes
1 answer
92 views

What is the formula that connects the average distance to the nearest point and the average number of points per unit volume?

We have an infinite 3D space with points randomly disseminated inside it. Let $p$ be the average number of points per unit volume. Let $d$ be the average distance to the nearest point. What is the ...
Omega Force's user avatar
0 votes
0 answers
45 views

Uniform Probability, Intuitive $X > Y$

Let's say $X$ is uniformly distributed from $0.3$ to $0.9$, and $Y$ is uniformly distributed from $0.6$ to $0.8$. Both IID. Is there any intuitive/geometric way of determining $P(X>Y)$? I know how ...
John Li's user avatar
  • 157
0 votes
0 answers
29 views

how to prove independency of 2 variables

The variable $(X,Y)$ is uniformly distributed over $$D=\{(x,y)\in\mathbb R^2:|x|+|y|\le1\}.$$ Let $$A=X-Y,B=X+Y.$$ Are $A$ and $B$ independent? I tried to prove $F(AB)=F(A)F(B)$, I tried to find the ...
Tomer's user avatar
  • 1
1 vote
1 answer
30 views

If U is uniformly distributed on [0, 1], is 2U - floor(2U) also?

If $U \sim U(0, 1)$, does $2U - \left\lfloor 2U\right\rfloor \sim U(0, 1)$? If $g(x) = 2x - \left\lfloor 2x\right\rfloor$, $\epsilon \in (0, 1)$, then $g(\frac{1}{2} - \epsilon) = 1 - 2\epsilon$. But $...
johnsmith's user avatar
  • 367
1 vote
1 answer
96 views

An upper bound for this probability $\mathbb{P}(U_1<U_2<...<U_j, U_j \in(U_{j+1},U_{j+1}+ \beta), U_{j+1}<U_{j+2}<...<U_k)$

Feel free to change the title. I want to prove that: $$\mathbb{P}(U_1<U_2<...<U_j, U_j \in(U_{j+1},U_{j+1}+ \beta), U_{j+1}<U_{j+2}<...<U_k) \leq \frac{\beta}{j!(k-j)!}$$ where $U_i$'...
stackQandA's user avatar
-4 votes
3 answers
157 views

Random vector $(X, Y)$ has a uniform distribution on the unit circle. [closed]

Faced with the following problem, I do not understand how to solve this problem: Random vector $(X, Y)$ has a uniform distribution on the unit circle. Will its components be independent? It is not ...
Bogdan Witcher's user avatar
0 votes
0 answers
28 views

On the convergence in probability of the maximum statistic of a random variable according to triangular and uniform

Set up Consider the example in section 2 of Ferguson (1982). Let $X_1, \ldots, X_n$ be i.i.d. with a distribution which with probability $\theta$ is the $U(-1, 1)$, and with probability $(1-\theta)$ ...
ytnb's user avatar
  • 600
0 votes
0 answers
40 views

Uniform distribution from Multivariate normal distribution [duplicate]

The following is from an optional exercise guide. The fact that the result should be a uniform distribution was given to me as a hint by a hasty professor, nevertheless I have been unable to solve it. ...
Lucas G's user avatar
  • 31
0 votes
0 answers
15 views

Calculating Expected Unique 2-Hop Neighbors in a Uniformly Distributed Network

In a 2D network with uniform node density $\rho$, each node has an average of $n_1 = \rho \pi R^2$ one-hop neighbors and $n_2 \leq 3 \rho \pi R^2$ two-hop neighbors. Each of these 1-hop neighbors ...
Khwrzm's user avatar
  • 1
0 votes
1 answer
33 views

Density of power with random variable

Random variables X, Y, Z are independent with uniform distribution from [0, 1]. find density of $XY^{Z}$ actually I'm stuck because power is a function (Random variable). I understand how to find ...
myfakeaccount's user avatar
1 vote
2 answers
329 views

Probability distribution of a random variable - Interview

this question is from a interview a friend had and I was curious how to solve it. The question is: I have two random variables, $a$ and $b$. $a=rand(1,100)$ $b=rand(a,100)$ The rand function ...
Analysis_Complex_Study's user avatar
1 vote
0 answers
42 views

Joint distribution of the first and second repeat in a sequence of random numbers

I have iid random variables $Z_1,Z_2,...$ that are uniformly distributed on the set $\{1,2,...,n\}$. I define $X(n)=\min\{k:Z_k=Z_j\text{ for some }j<k\}$ and $Y(n)=\min\{k:Z_k=Z_j,Z_i=Z_l\text{ ...
Marageku's user avatar
2 votes
0 answers
57 views

Probability that a random reduced fraction has an odd denominator

Suppose you choose a rational number at random and reduce it; what is the probability that the denominator is odd? I looked around and heard you can't define a uniform probability measure on the ...
Joseph Bendy's user avatar
0 votes
0 answers
22 views

Compute the number of days to collect $n$ collectables when receiving one collectable each day with uniform distribution. [duplicate]

Let $n$ different types of coupons. Jhon wishes to collect them all. Each day he receives one coupon with uniform distribution with complete independence between the days. Let $X$ be a random variable ...
MathStudent101's user avatar

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