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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0answers
39 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
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2answers
44 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
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2answers
27 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
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1answer
21 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
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0answers
15 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
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0answers
18 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
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1answer
38 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
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0answers
14 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
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0answers
23 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 = ...
0
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0answers
44 views

Chebyshev's Inequality and the length of a random vector

Suppose that we take $n$ iid random variables $X_1,\dotsc, X_n\sim \operatorname{Unif}[-1, 1] $ and define $Y_n=\lVert (X_1,\dotsc, X_n) ^\intercal \rVert $. By what means can we employ ...
0
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0answers
20 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
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2answers
21 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
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1answer
16 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
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1answer
41 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
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0answers
14 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
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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
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2answers
50 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
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1answer
17 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
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1answer
30 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
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2answers
79 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 ...
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3answers
40 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
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1answer
25 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 ...
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0answers
22 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: ...
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1answer
20 views

Find distribution function $F_Y(y)$ of random variable $Y$ [closed]

Tomorrow an midterm exam so I really need your help Let X be a random variable uniformly distributed in $[-1, 4]$. Say $Y = |X|$. Calculate the distribution function $F_y(y)$ of the random ...
0
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0answers
18 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
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2answers
44 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 ...
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1answer
48 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
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3answers
32 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
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0answers
17 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
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1answer
44 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
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0answers
7 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.
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0answers
13 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
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1answer
19 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 $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
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2answers
84 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)))?
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0answers
38 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
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3answers
30 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 ...
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0answers
12 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
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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 ...
3
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2answers
73 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?
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0answers
36 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 ...
1
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1answer
27 views

Shuffeled coin tossing

Had a small question: Let's consider the probability space $(\Omega, \mathfrak{F})=([0,1], \mathfrak{B})$ with Lebesgue measure $\mathbb{P}$, $\mathfrak{B}$ is Borel sigma algebra. Lets expand a ...
0
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2answers
45 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
1
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1answer
53 views

probability: nonlinear best predictor $\hat{Y} = g(X)$

Consider $X\sim\mathcal{U}(-1,1)$ and $Y = X^2$. The nonlinear predictor is defined as $$ \hat{Y} = g(X) = E_{Y|X}[Y|x_i] $$ Now $E_{Y|X}[Y|x_i] = \int_{-\infty}^{\infty}y\frac{f_{X, Y}(x, ...
0
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1answer
25 views

probability: best linear predictor $\hat{Y} = aX + b$

Let $X\sim\mathcal{U}(-1, 1)$ and $Y = X^2$. Since the best linear predictor is defined as $$ \hat{Y} = E_Y[Y] + \frac{\text{cov}(X, Y)}{\text{var}(X)}(x - E_X[X]) $$ Can I simple just write it as ...
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1answer
33 views

Determine probability of fewer than a certain number of events

Could anyone help with the following problem? My guts is telling me that the answer to part (a) is a normal distribution. Mainly, because I can't see where a uniform distribution would fit in this ...
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0answers
9 views

Using the rejection method to generate values

I'm trying to solve the following problem on rejection sampling: I think I have a good idea about what rectangle I should be using. In my mind, it would be a rectangle just large enough to encompass ...
1
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1answer
33 views

continuous probability: signal to noise ratio $\mu^2/\sigma^2$

$\DeclareMathOperator{\var}{var}\DeclareMathOperator{\cov}{cov}$ The signal-to-noise ratio (SNR) of a random variable quantifies the accuracy of a measurement of a physical quantity. It is defined ...
2
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1answer
44 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...
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0answers
38 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
1
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1answer
14 views

Find percentage of conforming items for normal and uniform distributions

I'm given the following problem: Could anyone give some insight into how to solve this? I understand that for the normal distribution, I will likely have to look the percentage up in a table. ...