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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1answer
18 views

unbiased estimator of $m^2$ of uniform distribution over $(0,m)$ [on hold]

I have a sample of size $1$ from a distribution that has $U\left(4,4+m\right)$ as probability density function. Is there any unbiased estimator for $m^{2}$?
2
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0answers
26 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
-2
votes
1answer
30 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
1
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1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
0
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1answer
53 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
0
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0answers
17 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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0answers
12 views

MLE for a two dimensional uniform distribution. [closed]

Take the case of a one dimensional uniform distribution, for e.g, X uniformly distributed on (-A,A) we have to such variables X and Y, X uniformly distributed on (-A, A) and Y uniformly distributed on ...
1
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2answers
29 views

Average distance from center of circle to evenly-distributed points within it

With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a ...
0
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0answers
43 views

Expectation of $v=\inf \{n\geq 2\,;\, X_n > X_1 \}$ when $(X_n)$ is i.i.d. uniform on (0,1)

Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...
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2answers
34 views

Maximum likelihood estimator on uniform distribution

I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me mad. Here it goes. Consider the uniform distribution on ...
1
vote
1answer
14 views

How random are numbers from geometric distribution

If I will choose a value $r$ as a random number from uniform distribution, I can be sure, that this value is totally random - because each value is equiprobable. However, what if I will take a value ...
1
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1answer
19 views

Maximum likelihood on uniform distribution

In a exercise i'm doing it is asked to find the maximum likelihood estimator of a random sample $X_{1}, ... , X_{n}$ of a population with distribution $X\sim U(- \theta , \theta) $. I've found that ...
2
votes
2answers
34 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
1
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1answer
26 views

Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at ...
1
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1answer
41 views

Distribution of function of a Random Variable

If $X$ is uniform on $(0,1)$, how would I go about finding the CDF of $Y=(X-X^2)^2$ ?Thanks.
1
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1answer
81 views

Uncertain about which probability method to use for the problem

Suppose I want to catch a bus (which runs every 10 minutes on average). What is the probability that: 1). You will wait for at least fifteen minutes before the bus arrives, and then, 2). 3 buses ...
0
votes
0answers
23 views

Conditional expectations with identical marginals and positively dependent but unknown joint distribution

Let $A$ and $B$ be random variables, each with marginal distribution $% U\left( 0,1\right) $, but unknown joint distribution $H\left( a,b\right) $. Suppose $A$ and $B$ are each stochastically (weakly) ...
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0answers
3 views

Sampling specific unit vectors

Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If $A = ...
0
votes
1answer
21 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
2
votes
0answers
33 views

Inconsistent answers with conditional expectations

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1]$. I want to compute: $$E[X+Y\mid\max\{X,Y\}≤(1/2)]$$ My first approach was the following. Let $X=\max\{X,Y\}$. ...
2
votes
2answers
61 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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2answers
49 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
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0answers
12 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
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1answer
37 views

Demonstrate uniform continuous distribution using tangible items?

What is the best way to explain "equally likely" in continuous uniform distribution to an audience using tangible or everyday items?
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2answers
45 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
2
votes
1answer
31 views

Stoppage time for sequence of uniform random numbers with a recursively shrinking domain

Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which ...
0
votes
1answer
26 views

How were the bounds of integration changed here in this probability problem on uniform distributions?

I have the following problem and solution: I don't understand how the bounds of integration were changed from 0 to 1, to $x^2$ to 1. I see where $1/\sqrt{y}$ was substituted in for $f(x|y)$ and 1 ...
1
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1answer
23 views

calculating probabilities about the sum of dependent discrete uniforms

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations ...
0
votes
2answers
44 views

Probability Question About Uniform Random Variables and Median

Let U, V, W ∼ Uniform(0, 1) be independent. Find the probability that the median (i.e., the second smallest) of these three random variables lies in the interval (1/4, 3/4). I cannot figure out what ...
0
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1answer
150 views

uniformly distributed random variables

Ram and Shyam wanted to meet at a park about 12.30 P.M.. If Ram arrives at a time uniformly distributed between 12.15 P.M. to 12.45 P.M. and if Shyam independently arrives at a time uniformly ...
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2answers
60 views

joint distribution, discrete and continuous random variables

This may be trivial, but if X is a random variable uniformly distributed over $[0,1]$ and Y is a discrete random variable such that $\mathbb{P} (Y=y_1) = \lambda \in (0,1]$ and $\mathbb{P} (Y=y_2) = 1 ...
2
votes
2answers
53 views

Mean of the difference between uniform random variables.

I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then ...
0
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1answer
45 views

interval learning unbiased estimator

Suppose that I draw $n$ points uniformly at random from $\mathcal{U}(-a,a)$ for some $a\in\mathbb{R}_+$. Denote this set of points $\{X_i\}_{i=1}^n$. Now for some $-a < x_1 < x_2 < a$, let a ...
2
votes
1answer
48 views

Indicator function and expectation

Consider a random variable $X$ uniformly distributed over $[0,1]$ and the indicator function $\mathbb{1}_{X \geq \tilde{x}} $ equal to one if $X \geq \tilde{x}$ and zero otherwise. We know that ...
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1answer
41 views

Conditional Probability Distribution for two Discrete Uniform Random Variables with given Correlation Coefficient

I consider a problem with two random variables $X, Y \sim Unif\{a,b\}$, for which I want to set a correlation coefficient $Corr(X,Y)=\rho$. Now, I am interested in the conditional probability mass ...
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2answers
39 views

How to prove that equation over probabilities has unique solultion or find counterexample?

Given equations: $$ \prod_{i=1}^n p_i = \prod_{i=1}^n (1-p_i)= \frac{1}{2^n} $$ where $p_i\in (0,1), i=\overline{1,n}$. Is it true that this system has unique solution $p_1=p_2=\ldots=p_n=\frac12$ ...
0
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4answers
71 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that ...
2
votes
1answer
32 views

Calculating power of a Hypothesis Testing Problem based on Uniform distribution

Consider the problem of testing $H_0:a=0$ against $H_1:a=1/2$ based on a single observation X from U(a,a+1). The power of the test "Reject $H_0$ if $X>2/3$" is (A)1/6 (B)5/6 (C)1/3 (D)2/3 ...
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0answers
35 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
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0answers
34 views

Continuous Uniform Distribution

Suppose $X$ follows a continuous uniform distribution from 1 to 5. Determine the conditional probability $P(X > 2.5 | X \le 4)$ I am not sure I know how to do ...
0
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1answer
29 views

Order statistics and biased estimators

Would someone be able to check my work on this question: Suppose that $Y_1,Y_2,...,Y_5$ is a random sample from a uniform distribution over the interval (0, theta). Determine if the followng ...
3
votes
1answer
98 views

Expectation of maximum of two independent random variable with known distribution [closed]

Assume $X$ and $Y$ are two random variables such that $X\sim \textrm{Unif}(0,1)$ and $Y=e^{-t}\times a $ where $t\sim \mathrm{Exp}(\lambda)$ and $a\sim \textrm{Unif}(0,1)$. What is ...
4
votes
2answers
99 views

Sum of discrete and continuos random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are ...
2
votes
2answers
85 views

Two people meeting, expected time of waiting

$A$ and $B$ are supposed to meet. $A$ arrives in a randomly chosen (uniform distribution) moment between $2$ and $3$ pm. $ B$ arrives at $2$ pm with probability equal to $0,5$ and in a randomly ...
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votes
0answers
27 views

Separating a random variable

Let α and β be random variables uniformly distributed from 0 to 1. Let λ= k1α -c1α + k2β -c2β. Let x be the random variable that is uniformly between ...
0
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1answer
25 views

Expected length of shortest interval containing numbers drawn at random

A random idea: If you draw $n$ numbers uniformly at random from $[0,1]$, what is the expected length $L_n$ of the shortest interval that contains all but one of them? Clearly, we have $$L_2 ...
0
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0answers
63 views

Relationships between Uniform and Pareto Distributions

If $X$ is uniformly distributed over $(a,b)$ and $Y$ is pareto distributed with parameters $(min,c)$, what is the distribution of Z in the following cases? (a) $Z = X + Y$ (b) $Z = XY$ (c) $Z = ...
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0answers
56 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...
2
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0answers
67 views

Question on uniform distribution of points on a sphere.

Let N points be uniformly distributed on the surface of a unit sphere $S^2$. What is the probability that every spherical cap of area A contains at least one point? The area $A$ depending on the ...
0
votes
1answer
29 views

Uniform Distribution $Y:=(X+1)^2$

$X \sim uniform[0,4]$ Another probability variable $Y$ is defined as $Y:=(X+1)^2$. I'm searching for the CDF of $Y$. Thing's I already know: If $W:=X+1$ then $W \sim uniform[1,5]$ $Y=W^2$, so ...