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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17 views

Negative exponential of an exponential random variable is a uniform random variable?

I know that the negative log of a uniform random variable is an exponential random variable. I am trying to prove the reverse, but I seem to have arrived at something that doesn't make sense. Define $...
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1answer
28 views

Uniform random variables question

Let U and V be independent random variables, both uniformly distributed on [0, 1]. Find the probability that the quadratic equation $x^ 2 + 2Ux + V = 0$ has two real solutions. My solution: The ...
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0answers
52 views

${α⋅ \log(n)}$ is not uniformly distributed mod1 in $[0,1]$

$\qquad \qquad \bbox[15px,border:2px solid red] { x_n:=\text{\{α$\cdot$ log(n)\}}_{n\in \mathbb N}}$ I want to show that the sequence $x_n$ is not uniformly distributed mod1 in $[0, 1]$ for any $α\in ...
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2answers
67 views

Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$

Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$ where $X_{(n)} = \max_i X_i$. It was suggested to use Basu's Theorem which I am ...
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2answers
38 views

Probability that the bus will arrive within 4-5 minutes

The bus runs at intervals of 10 minutes, and at a random moment you come to a stop. What is the probability that the bus will arrive within 4-5 minutes? My textbook says the answer is $\frac{1}{10}$ ...
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0answers
40 views

moment generating function uniform distribution

I'm trying to prove a simple example of MGF for $E[x]$ for a simple RV with uniform distribution over range $[0,10]$ but not sure where I'm getting stuck: $A(t) = E[e^{tx}] = \int_{0,10}e^{tx}dx/10$ *...
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1answer
25 views

Entropy of discrete and continuous uniform distributions

Despite a similar post here, I read that the entropy of a uniformly distributed discrete random variable is always log base $2$ of the number of observations in the dataset, $H(X) = \log(N)$. Is this ...
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1answer
42 views

Suppose $U \sim Unif(0,1)$ and $Z \sim Unif(U,3+U)$. How can I find the pdf for $U + Z$?

Suppose $U \sim Unif(0,1)$ and $Z\mid U \sim Unif(U,3+U)$. I would like to find the pdf for $U + Z$, which in my process on $Z$ is a continuation of $U$. Is there a straightforward way to derive this?
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3answers
36 views

Finding the joint distribution of two independent uniform random variables $X,Y$ given event $E$

If $E = \{\text{either} \ X < 1/3 \ \text{or} \ Y<1/3\}$ for $X\sim Unif(0,1)$ and $Y\sim Unif(0,2)$, does the joint distribution of $X,Y$ given event $E$ exist? I am assuming that $X$ and $Y$ ...
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1answer
27 views

What is the pdf for a jointly uniform distribution inside a triangle?

I have a triangle bounded by $0 \leq x, y \leq 1$ and $x + y \geq 1.5$. I'm told that points are uniformly distributed within this triangle. I am wondering how I can find the pdf? Is it simply solving ...
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1answer
36 views

How to find the distribution function

Random variable X is uniformly distributed over the interval [0,1]. $Y = -ln(X)$.I need to find the distribution function for $Y$. So i did this: $$P(Y \leq x)= P(-ln(X) \leq x) = P(ln(X) \geq -x) = 1 ...
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1answer
15 views

Find the intervals for the distribution function

Random variable X is uniformly distributed over the interval [0,1]. $Y = X^{2}$.I need to find the distribution function for $Y$. So i did this: $$P(Y \leq x)= P(X^{2} \leq x) = P(X \leq \sqrt{x}) = F(...
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1answer
42 views

Probability two uniform distribution(0,1) = 2/9

Two numbers are independently and uniformly chosen from the interval (0,1). What is the probability that the sum of the numbers is less than 1 and the product of the numbers is less than 2/9? (Note ...
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2answers
66 views

What is the pdf of $\frac{|x-y|}{(x+y)(2-x-y)}$ when $x,y$ are i.i.d uniform on $[0,1]$?

If $x,y$ are i.i.d uniform random variables on $[0,1]$. I know that the PDF of $|x-y|$ is: $$f(z) = \begin{cases} 2(1-z) & \text{for $0 < z < 1$} \\ 0 & \text{otherwise.} \end{cases}$$ I ...
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1answer
23 views

Uniform distribution and conditional distribution

If $U \sim Uni([0,1])$, does $U| U\ge 0.5 \sim Uni([0.5,1])?$ If it's false, why? and if it's true, can it be generalized for any Uniform Distribution?
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84 views

If $U$ is uniformly distributed on $S^{d-1} \subset \mathbb{R}^d$, what's the distribution of its orthogonal projection onto any vector?

Let $U \in S^{d-1} \subset \mathbb{R}^d$ follow a uniform distribution on a sphere. Let $v \in \mathbb{R}^d.$ Then is the orthogonal projection $U^{T}v=\langle U,v \rangle$ uniformly distributed, and ...
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1answer
24 views

Find pdf of transformation of two random variables using CDF [duplicate]

Let $X,Y \sim$ Uniform$(0,1)$ be independent. Find the PDF for $X/Y$. Let $Z=X/Y$. We want to find $F_z(z)=P(Z \leq z)=P(X/Y \leq z)$. We can make $Y$ super small with fixed $X$, and conversely we ...
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3answers
48 views

Probability that $\max(X_1, \ldots, X_n) - \min(X_1, \ldots, X_n) \leq 0.5$

Consider $n$ IID random variables $X_1, \ldots, X_n \sim U(0,1)$. What is the probability that $\max(X_1, \ldots, X_n) - \min(X_1, \ldots, X_n) \leq 0.5$. Denote $Z_1, Z_n$ as the min and max ...
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2answers
21 views

arctan of ratio of two normal variables is uniform

Say $X, Y$ are independent standard normals, and $\theta = \arctan(Y/X)$. Prove that $\theta$ is uniformly distributed over it's range. It is pretty intuitive that the distribution of $\theta$ would ...
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0answers
25 views

Probability of $A_i \leq B_j$ given $A$ and $B$ sorted lists of integers

I am looking for a solution to the following problem: Suppose we randomly select $k$ integers from $[0,$ $2^b - 1]$. We then sort each $k/2$ in non-decreasing order and store in two lists $A = [a_0, ...
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0answers
31 views

Distribution of x, picking r and theta uniformly

Say I generate a random $R$ and $\Theta$ (each uniformly), where $R$ is in $[0, R_0]$ and $\Theta$ is in $[-\pi, \pi]$. I create a vector from it, i.e. let $X=Rcos(\Theta)$ (and $Y=Rsin(\Theta)$, but ...
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0answers
25 views

Sum of two Uniform random variables defined on arbitrary intervals

I derived this distribution, but I'm not entirely sure of the middle point. Intuitively it should be $\frac{a+b+c+d}{2}$. For two independent Uniform rvs, $X \sim R[a,b], Y \sim R[c,d]$ derive the ...
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1answer
70 views

When is the sum of two uniform random variables uniform?

Suppose that $X$ and $Y$, two random variables, are both uniformly distributed over $[0,1]$. Let $Z=\frac{1}{2}X+\frac{1}{2}Y$. I know that in general, $Z$ is not uniform. For instance, $Z$ is not ...
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1answer
25 views

Joint probability for 2 uniform distributions

X and Y are independent and uniformly distributed on the interval (0, 1). If U = X + Y , and V =X/Y find the joint density for U and V and the marginal densities for U and V. Given that $$f_{UV}(uv) =...
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1answer
29 views

Product distribution of two uniform distributions which are centered around 1

Consider the product distribution $Z = X_1\cdot X_2$ for $$ \begin{aligned} X_1 &\sim \textrm{Uniform}[1 - a, 1 + a] \quad, \quad 0 < a < 1 \\ X_2 &\sim \textrm{Uniform}[1 - b, 1 + b]...
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1answer
21 views

Is the set of differences of independent uniform random variables, independent?

Consider a finite set of uniformly distributed, independent random variables $\mathbf{X} = \{X_1, X_2, \dots, X_n\}$ on the unit interval. The absolute values of the difference between these variables ...
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2answers
65 views

What is the probability to form a triangle with the three pieces of the stick?

On a stick $1$ meter long is casually marked a point $X \sim U[0,1]$. Let $X=x$, is also marked a second point $Y\sim U[x,1]$. 1) Find the density of $(X,Y)$ showing the domain. $$\rightarrow \quad ...
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1answer
21 views

Cumulant generating function of continuous uniform distribution

With $f(x)=\frac{1}{b-a}$, I got the MGF, $M(t) = \frac{e^{bt}-e^{at}}{t(b-a)}$, and the cumulant generating function, $K(t) = \ln (e^{bt}-e^{at}) - \ln [t(b-a)]$. However, when I tried to find the ...
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1answer
25 views

Find a probability given specific cdf values

Given $X$ is uniform random variable, $P\{X>1\} = 0.6$ and $F(2) = 0.5$. Find $P\{-1\leq X < 3\}$. My solution is: $P\{X>1\} = F(\infty) - F(1) = 0.6$. So $F(1) = 0.4$ And now I assume that ...
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1answer
29 views

Showing $X_{(n)}$ is not complete for $\theta \in [1,\infty)$ when $X_i$'s are i.i.d $\text{Unif}(0,\theta)$

I am trying to show that the order statistic $X_{(n)}$ for a set of RV $\{X_i\}_{1}^{n}$ where $X_i\overset{iid}\sim \text{Unif}(0,\theta)$ is complete when $\theta \in (0,\infty)$ but not when $\...
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1answer
29 views

2 Random variables that are connected to one another

Let $A$ be a random variable distributed uniformly on $(0,1)$. Given $A=a$ , the random variable $B$ is binomial with: $B \sim \text{Bin}(n=5, p=a)$ (1) Find the PDF of $A$ given $B=b$. $f_{A | B=b}(k)...
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2answers
31 views

Expectation of a equation with uniformly distributed variable

There is the following problem: One maximises profit (v-0.5)x, where x ∈ {0,1} and v~U(0,1). Now, in order to maximise this one chooses x=0 if v<0.5, and x=1 if v≥0.5. Now, one has to find the ...
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1answer
22 views

Expectancy of choosing a number and then choosing another number depending on the first choice

You randomly choose a number from this set of $n$ numbers: $\{1, 2, \dots , n\}$. Let $X$ be the number you chose. Let $Y$ be a natural number that is chosen randomly uniformly (discrete) from this ...
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1answer
53 views

Uniform Distribution in a geometric Shape

$(X,Y)$ distributed uniformly on a triangle which its vertices are: $(-1,0), (0,1), (1,0) $ Let $x$ be a constant number, and assume $ x \in (-1,1)$ Calculate: $\mathbb{P}(Y \geq 0.5 | X=x)$ $\...
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1answer
20 views

Interpretation of Simple Uniform Marginal Density Example from Mathematical Statistics by Rice

I am stuck trying to visualize Example B from Mathematical Statistics and Data Analysis 3rd ed by Rice. The examples revolve around the concept of independence of Random Variables which is defined in ...
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0answers
18 views

Balanced distribution of resources

Let's say we have 9 storage units with the following current quantities: 152, 153, 154, 159, 157, 147, 140, 265, 205. The 9 storage units will be supplied daily, ...
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89 views

Find the PDF of a random vector

So I have a question in probability theory that's driving me insane. I know it is easy but I can't seem to wrap my head around it. Assume we have $U_1\sim U[-1,1]$ and $U_2 \sim U[0,2]$ which are two ...
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2answers
60 views

Finding maximum likelihood estimator of $\theta$

Let $X_1, \ldots, X_n$ be independent random variables with density $$ f(x;\theta) =\begin{cases} \frac{1}{2i\theta} &, -i(\theta-1)\le x \le i(\theta+1) \\ 0&, \text{ elsewhere } \end{cases} $...
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1answer
54 views

$(X,Y)$ on a triangle

Let $(X,Y)$ a random variable uniformly distributed on the triangle $(0,0)$, $(0,1)$, $(1,0)$. Find the density of $(X,Y)$. $\rightarrow f_{X,Y}(x,y)=2$ Determine if $X$ and $Y$ are independent or ...
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1answer
25 views

Probability question - Normal and Uniform

In a factory there are $2$ machines that create tubes (They are independent from each other) Length of the tubes of machine A is distributed normally with an expectancy of $101$ cm. and a Variance of ...
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2answers
67 views

Find $\mathbb{P}(A\cap B^c)$ where $A=\{X_1+X_2<1\}$ and $B=\{X_1+X_2+X_3<1\}$

Let $(X_1,X_2,X_3)$ three independent random variables with uniform distribution $[0,1]$. Let $A=(X_1+X_2<1)$. Find $\mathbb{P}(A)$. $\rightarrow \mathbb{P}(A)=\int_{0}^{1}[\int_{1-x_2}^{1}dx_1]...
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2answers
42 views

Probability of a Uniform Random Variable [closed]

Let $X_1, X_2, X_3$ be iid Uniform (0,1) random variables. How do I find the probability that $X_{\min} = \min[X_1,X_2,X_3]$, is between 0 and 1/2?
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0answers
29 views

Expected Euclidean Distance [duplicate]

Let $ S = \{(x,y) \in \mathbb R^2 : x^2+y^2 < 1\}$ be the unit circle in $\mathbb R^2$. Let $(X_1, Y_1), (X_2, Y_2),$ be independent, both having uniform distribution over $S$. Let $D$ denote the ...
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1answer
27 views

$U \sim U(0,1)$ and let $X$ be the root of $3t^2 −2t^3 −U = 0.$Show that $X$ has p.d.f. $f(x)=6x(1−x)$, if $0\leq x\leq 1$

Let $U \sim U(0,1)$ and let $X$ be the root of the equation $3t^2 −2t^3 −U = 0.$ Show that $X$ has p.d.f. $f(x)=6x(1−x)$, if $0\leq x\leq 1,\;=0$,otherwise. I really don't know how to begin solving ...
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41 views

Let $Y = X − [X]$, where $X\sim U(0,\theta)$. Show that $Y \sim U (0, 1)$

Let $X \sim U(0,\theta)$, where $\theta$ is a positive integer. Let $Y = X − [X]$, where $[x]$ is the largest integer $≤ x$. Show that $Y \sim U (0, 1)$ Clearly, the support of $Y$ is $S_Y = [0,1]$. ...
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1answer
87 views

KL-Divergence of Uniform distributions

Having $P=Unif[0,\theta_1]$ and $Q=Unif[0,\theta_2]$ where $0<\theta_1<\theta_2$ I would like to calculate the KL divergence $KL(P,Q)=?$ I know the uniform pdf: $\frac{1}{b-a}$ and that the ...
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2answers
33 views

Probability that maximum and minimum of two segments is less than a given value

I'm working on probability theory and recently I've got stuck with the following task: Random point $A$ divides the segment $[0, 1]$ on two segments. Let $M$ be the size of biggest segment and $m$ be ...
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1answer
21 views

Expectation of differences between arcs on a circle

Consider a circle with a circumference of $n$. On this circle, I define two arcs of length $k<n$, $A_1$ and $A_2$. The centres of the two arcs are uniformly distributed on the circle. Let $\Omega_{...
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1answer
23 views

Derived distribution of a function of two random variables

I am trying to derive the following problem by myself from Bertsekas "Introduction to probability" I cannot get the CDF of the 2nd interval (Z < 1) correctly. please identify my error. ...
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1answer
24 views

Set of random numbers with uniform distribution - justify the distribution of the differences

Make a set of random integers, uniformly distributed between 0 and n=10^4. The size of the set is n^2. ...

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