For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

learn more… | top users | synonyms

2
votes
1answer
76 views

Probability of average distance from origin of unit circle less than half

Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than half?
2
votes
1answer
266 views

Show that there is no discrete uniform distribution on N.

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?
2
votes
2answers
54 views

The Hurried Duelers brainteaser

This question is similar as this other one asked in the forum, but I am trying to give it a different twist. Unfortunately, I am not getting to the same answer, so there might be something wrong in my ...
1
vote
1answer
29 views

Determine the Hypergeometric probability function using sample space in which the selection is ordered

I'm unable to think through this question please help. Suppose our sample space distinguishes points with different orders of selection. For example suppose that $S =\{SSSSFFF\ldots\}$ consists of ...
2
votes
2answers
86 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
votes
1answer
27 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
3
votes
1answer
61 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
5
votes
1answer
48 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
vote
0answers
28 views

sum of random variables?

$x\to$ uniformly distributed on $(0,1)$ $y\to$ uniformly distributed on $(0,2)$ $z\to$ uniformly distributed on $(0,4)$ What is the probability that $2x+3y < z$? I tried to do it geometrically ...
1
vote
1answer
40 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
1
vote
1answer
27 views

expectation of uniformly distributed $n$ number of samples

I am trying to fine the expectation: $E((x_1+ x_2+ \cdots +x_n )^2)$ as a function of $n$ where all $x_1$ to $x_n$ have uniform distribution $U(0,1)$. I can do if there is only $x_1$ and $x_2$ but ...
0
votes
1answer
13 views

Comparing Uniform Random Variable

X,Y and Z are uniformly distributed random variable on (0,1) What is the probability that X+Y>Z? I tried to do it geometrically and find the volume x+y in the given limits ie 0 to 1 for both x and y. ...
-6
votes
2answers
115 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...
0
votes
2answers
56 views

How to find $\lim_{n\to \infty} P(a≤(X_1X_2…X_n)^{-n/2}e^{n/2}≤b)$ where $X_1,X_2,…,X_n \sim U[0,1]$?

I am trying to calculate $$\lim_{n\to \infty} P(a≤(X_1X_2...X_n)^{-n/2}e^{n/2}≤b)$$ in terms of $a,b$, where $$X_1,X_2,...,X_n \sim U[0,1]\,\,\,\,\,\,\,(i.i.d.)$$ and $$0≤a<b$$ My attempt is to ...
-1
votes
2answers
54 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
0
votes
1answer
28 views

If the side length of a square follows uniform distribution, how to find the mean and variance of its area?

A square has side of length $X$ cm, where $X\sim U[4,10]$. Find the mean and variance of the area of the square. I understand how to get the mean and variance for the length of each side, but simply ...
0
votes
0answers
24 views

The distribution of the sum of a uniform random variable and a binomial random variable

I'm asked to find the distribution of $U=X+Z$, where $X\widetilde~R(0,1)$ - That is, $X$ has a uniform distribution for $x\in]0;1[$ $Z\widetilde~bin(1,1/2)$ - That is, $Z$ has a binomial ...
1
vote
0answers
21 views

uniform distribution probability and mle

For part a), isn't the probability = 1? And I'm not sure what happens as $n\rightarrow\infty$; isn't the probability 1 also?
0
votes
1answer
46 views

Why does the MLE of the uniform distribution not satisfy a Central Limit Theorem?

For $X ~ U(0,\theta$) The MLE of $\theta = \max{x_i}$. Why does this not satisfy $\sqrt{n*I(\theta)} *( \max(x_i) - \theta) -> Z $ Where Z has a normal distribution? I understand that $\max{x_i} ...
1
vote
0answers
30 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
1
vote
0answers
36 views

Obtain distribution of mid-range in uniform

I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation. $M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = ...
1
vote
1answer
44 views

Conditional distribution on the unit circle and a square

Let (X,Y) be uniformly distributed over $B=\{(x,y) \in \mathbb{R}^2: x^2+y^2 \leq 1 \}$ resp. $Q=[-1,1]^2$. Now I want to calculate the conditional distributions and of Y given X=x. And then the ...
1
vote
2answers
41 views

Uniform distribution with unknown interval boundary

Suppose that a random variable $Y$ is uniformly distributed on the interval $[-a,a]$, with $a > 0$. Suppose that the random variable $X$ is uniformly distributed on the (stochastic) interval ...
0
votes
1answer
20 views

Calculate $P(A_n)$, where $P$ uniform distribution on $[0,1]$

$P$ uniform distribution on $[0,1]$. $$A_n=\bigcup_{i=1}^{2^n-1} \left [ \frac{2i-1}{2^n}, \frac{2i}{2^n} \right ], n \in \mathbb{N}$$ To calculate $P(A_n)$ do we have to do the following?? ...
1
vote
1answer
57 views

Distribution of differences between adjacent sorted uniform random variables on $[0,1]$

I saw this question on Mathematica.stackexchange, and I wonder what distribution the answer gives. Asymmetric definition Let $(X_1,X_2,\ldots,X_{n-1})\sim$ i.i.d. $U[0,1]$, and ...
0
votes
0answers
21 views

Universal hash function when size of hash is p^m

Can we define universal hash function from $U \rightarrow T$ when $T=\{0,1,2,..,m-1\}$ and $m=p^a$? (where $p$ is a prime and a is an integer) I know that we can define universal hash funciton when ...
3
votes
0answers
105 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
0
votes
2answers
55 views

Random variables and Linearity

I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a ...
1
vote
1answer
21 views

Number of uniform hash functions

how many uniform hash function I can create when I want to hash elements from $U$ where $|U|=m \cdot r$ , $m,r$ are integers. a hash function $h:U \rightarrow T $ , $|T|=n$ is uniform if ...
2
votes
1answer
35 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
4
votes
2answers
87 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
1
vote
3answers
46 views

Biggest among three uniform random variables

Question: Given three random variables $X, Y, Z$ of independent uniform distribution in range [0,1], what's the probability for $X$ to be the biggest one? I've come up with two solutions but they ...
0
votes
1answer
36 views

Resultant mean and variance of gaussian distribution

X be random Gaussian variable with mean u1 and variance v1. u1 itself is a random variable which is also gaussian distributed with mean u2 and variance v2. Then the distribution of X will be ...
0
votes
0answers
38 views

Finding the limiting distribution $n\min(X_1, \dots , X_n)$ with uniformly distributed $X_i$

Find the limiting distribution of $nY_n$ where $Y_n = \min(X_1, ..., X_n)$ and $X_1, ..., X_n\sim \operatorname{unif}(0,2)$ are uniformly distributed random variables. Here is what I did: ...
0
votes
0answers
24 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...
2
votes
2answers
55 views

Why is this true? (sum of 2 uniform distributions)

If $X\sim U[0,1]$ and $Y\sim U[-1,0]$ and they are independent, then the distribution of $X+Y$ is not simply $U\sim [-1, 1]$, but it is the sum of 2 independent $U\sim [-0.5 ,0.5]$ distributions. Why ...
-1
votes
1answer
66 views

Cdf and Pdf of independent random variables(iid)

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
1
vote
3answers
37 views

Probability uniform with transformation

Given $X,Y$ being discrete random variables that are independent and can take on values $[0,1,\dots,N]$ with equal probability, what is the distribution of $\max[X,Y]=Z$? Or any other transformation ...
0
votes
0answers
20 views

How to rewrite function for squared uniform distribution

The question is as follows: I am evaluating the following integral: $$\int_o^1\frac{\exp(\sqrt{1-x^2})}{\sqrt{x}}dx$$ by assuming it equals $E[f(U)]$ for a uniform distribution. I worked it out via ...
0
votes
1answer
47 views

What is the centroid of $x^2+{(-y^3+1)}^{2/3}=1$

I am wondering what exactly is the centroid of $x^2+{(-y^3+1)}^{2/3}=1$. It is a closed implicit shape. I want to know if solving for the centroid is the same thing as solving for a point with the ...
0
votes
0answers
11 views

Dinamic programming for sum of two largest values from a Uniform parent

I must find a recursion formula to find the sum of two largest values from a Uniform parent [0,1]. I need dinamic programming but I don't know how to organize it.
0
votes
0answers
51 views

Generate random numbers with beta distribution from uniform distribution

How can I generate a series of random numbers with beta distribution from random numbers with uniform distribution? I am aware that using inverse transformation method is at least very difficult or ...
0
votes
1answer
20 views

Finding a measurable function with an independent uniform distribution

Suppose $X,Y,U$ are random variables on some probability space such that $U$ is independent of $(X,Y)$. Prove there exists a measurable function $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
0
votes
2answers
98 views

Expected value of the sum of the two largest values from a Uniform parent

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?
1
vote
0answers
67 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
0
votes
3answers
35 views

Find PDF of the random variable Y. Please help!

Let $X_1$ and $X_2$ be independent and identically distributed Uniform $(0,1)$ random variables. Let $Y = \max(X_1, X_2)$. Find the PDF of the random variable $Y$. I am having a hard time progressing ...
1
vote
0answers
13 views

Even dirstibution of a small set of random choices into a small set of buckets

Is there a way to evenly distribute randomly selected small set of items from a relatively larger set into to a small number of buckets using a hash function? For ex: Randomly select 20 numbers from ...
1
vote
0answers
29 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
4
votes
2answers
160 views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
0
votes
0answers
53 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...