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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34 views

Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
0
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1answer
29 views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = (\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{...
2
votes
1answer
40 views

Showing that if $X \sim \operatorname{Exp}(1)$, then $Y = F_X(X)$ has uniform distribution on $[0,1]$

Let $X \sim \operatorname{Exp}(1)$, and show $Y = F_X(X)$ has uniform distribution on $[0,1]$. I calculated $F_Y$, since the cumulative distribution function identifies a distribution. We have: $$\...
0
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1answer
62 views

Uniform Distribution / Normal Distribution

Let the random variable X ~ U ( 0, k ) and Y is a second random variable such as Y | X ~ N ( X , 1). a) Determine the Y density function if k = A . b) Determine the value of k if COV [X , Y ] = B. a) ...
0
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1answer
27 views

Why these two distributions are different when I calculated them?

Two insurers provide bids on an insurance policy to a large company. The bids must be between 2000 and 2200 . The company decides to accept the lower bid if the two bids differ by 20 or more. ...
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0answers
51 views

Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...
2
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2answers
30 views

how far the distribution from the uniform distribution

I have two discrete probability distributions $P$ and $Q$, where $P=(p_1,...,p_n)$ and $Q=(q_1,...,q_n)$, in addition I have uniform distribution $U=(\frac{1}{n},...,\frac{1}{n})$. The question is ...
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1answer
33 views

Probability involving the maximum of i.i.d. uniform r.v.'s

The question is : $100$ numbers are independently and uniformly distributed on $(0,1)$.Then what is the probability that the maximum of these numbers will be at most $0.9$? How can I solve it? ...
1
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2answers
56 views

estimate a probability

Let $X_1....X_{48}$ be independent random variables, each follows a uniform probability distribution over [0,1]. What is the best way to estimate P($\Sigma_{i=1}^{48} X_i > 20)$?
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2answers
36 views

Transformation of the uniform distribution

I struggle to understand the transformation of a random variable with uniform distribution. For example: Let $X\sim \text{Uniform}(0,1)$ and $T=-2\ln(X)$ and I want to find the CDF of $T$, then I ...
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3answers
44 views

How could I find the covariance for $X$ and $Y$ in this case?

If $X \sim U(-1, 1)$ (so $X$ is uniformly distributed between $-1$ and $1$) and $Y = X^2$, what is the covariance between $X$ and $Y$? Are they independent? So the formula for covariance is: $\...
3
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1answer
26 views

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn't find the mean and variance of the order statistics of a discrete ...
0
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2answers
25 views

Notation inconsistency? Standard uniform distribution U(0,1)

Very simple and quick question. Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance, for example for normal distribution: $$N(0,1)...
0
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1answer
29 views

Uniformly Choosing a number from a range [closed]

May you please help me how I can choose uniformly a number from a range. I have to use this for trust evaluation in social networks such as the following clause: Each user has a quality ...
1
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1answer
33 views

Probability that n points on a circle are in one quadrant

Question Points $A$,$B$ and $C$ are randomly chosen from a circle, What is the probability that all the points are in one quadrant ($\frac{1}{4}$ circle)? My try Using this answer about ...
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0answers
12 views

Estimation of copulas

For estimation of a parameter of bivariate copula, we really need bivarite data? or from two different one dimensional data we can estimate copula parameter?
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1answer
35 views

How do I evenly distribute some time? [closed]

Here's my problem. I have a device (ozone generator) which produces 10g of ozone per hour, or 2.77778 milligrams per second (I think). I need to be able to control the production per hour by pulsing ...
-1
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2answers
43 views

Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
0
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1answer
34 views

Probablity of normal distribution when x is a function

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is ...
0
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1answer
24 views

Method to find out how distributed are a certain set of data?

Assume I have array of $A_{3\times120}$ Each row of matrix A corresponds to a shape which is generated by its three row elements as below: $r=1+a_1\cos(\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)$ ...
1
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1answer
46 views

Probability that $U_1 \geq U$, $U_2 \geq U$, $U_3 < U$, $U_4 \geq U$, $U_5 < U$, $U_6 \geq U$, $U_7 \geq U$, for i.i.d. uniform $U_k$s and $U$

Let $U,U_1,U_2,...$ be independant, on [0,1] uniform distributed random variables. Let $E$ := {$U_1 \geq U,U_2 \geq U,U_3 < U,U_4 \geq U, U_5 < U,U_6 \geq U,U_7 \geq U$}. Find the probabiliy $...
1
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1answer
27 views

Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
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3answers
40 views

Distribution of ages of 3 children in a family

Please consider the following problem: A family has 3 children, creatively named A,B, and C. (a) Discuss intuitively (but clearly) whether the event “A is older than B” is independent of the event “...
6
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1answer
88 views

Order statistics for discrete uniform random variables

Let $X_i, i=1,\cdots,N$ be i.i.d. discrete uniform random variables, taking values in the range $\{0,1,...,M-1\}$. Let $X_{(i)}$ denote the $i$-th order statistic. What are the values of $\...
0
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1answer
27 views

Probability that a set of uniformly distributed random variables is 'greater' than another such set.

Suppose we generate several uniformly distributed random variables (between 0 and 1), and arrange them in descending order to form a set [A1, B1, C1...]. We then do the same process to form a second ...
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2answers
27 views

Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$ Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases} $$ ...
0
votes
1answer
40 views

Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$

Suppose $X_1, X_2, ..., X_n$ are independent uniformly distributed random variables on [0,1]. Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$ I've ...
1
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1answer
48 views

Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$

Suppose $X_1, X_2, ... X_n$ are independent and uniformly distributed (on $[0,1]$) random variables. Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$ My thoughts were the ...
0
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1answer
29 views

Distribution function of uniform distribution

I don't know why the distribution of this question is that when x is in between 0 and theta. In solution.. is that right? I searched the distribution of uniform distribution. But it is alike with that ...
0
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0answers
34 views

Joint pdf of two uniform random variables on a unit line segment

Let $X$ be a standard uniform random variable, define $Y=1-X$. Then supposedly $X$ and $Y$ are uniform over a 1-simplex, so their joint distribution should be Dirichlet of order $K=2$, and $\alpha_1=\...
1
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2answers
21 views

Integration Question Arising from Attempting to Compute a Marginal Distribution

I have a question about the last step of a proof on page 37 of All of Statistics. The entire proof is here for sake of completeness, but I don't think grokking it in its entirety is necessary for ...
1
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1answer
24 views

Mathematical formula for equal distribution of amount among different group [closed]

Please let me know if you think i should edit my question or description. Problem statement: lets say i have spent $x on a sports material which needs to shared among total y memeber of the team. But ...
5
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0answers
32 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
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0answers
31 views

How to define a uniform probability distribution over a convex polytope / polyhedra and add them?

Let $P$ be a convex 3d polyhedra / 2d polytope constrained by a set of linear inequalities $Ax<= b$. 1.How to define a uniform probability distribution over a polytope/polyhedra? Let us say we ...
1
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1answer
60 views

is arcsin() evenly distributed?

Given a uniform random distribution P of real numbers from [0,1] how might I prove (or disprove) that the map from P to Q of $(p\in{P} \rightarrow q=arcsin(p)\in{Q}) $ is a uniform distribution over [...
1
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1answer
18 views

Probability of not choosing from a set of replaced values

The problem statement is as follows: There is a set of numbers N numbers $1..N$ (eg: N = $10^6$) N numbers are chosen uniformly and independently with replacement I would like to be able to ...
2
votes
2answers
25 views

Getting the marginal distribution from the joint pdf

To bein with, I did the following calculations: $$ Y\sim Uniform(0,x)\\ f_x(x)=\{\frac{1}{x^2},x\ge1\}\\ f_{y|x}(y)=\{\frac{1}{x},0\le y \le x\}\\ f(x,y)=f_x(x)f_{y|x}(y)=\frac{1}{x^3},x\ge 1,0\le y\...
1
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1answer
37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
4
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2answers
61 views

Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
0
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0answers
33 views

Find continuous stochastic variable $X$ with PDF $f_X = \frac{1}{x^2}$

Given the uniform stochastic variable $U$ defined on the interval [0,1]. Using $U$, define a continuous stochastic variable $X$ with probability density function (PDF) $$f_X(x) = \begin{cases} \frac{1}...
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2answers
26 views

Find function $h$ so that $h(U,V)$ equals density of $f(a) da$ for $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be independant and uniform distributed on [0,1]. Now I want to find a function $h$ so that $h(U,V)$ is equal to the density $f(a)...
2
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1answer
29 views

Computing the distribution of a uniform r.v. with parameter being another uniform r.v.

I have this: Let $X\sim U(0,1)$, $Y\sim U(X,1)$. What is the distribution of variable $Y$? My answer: I use a geometric approach since everything happens in the square $(0,1)\times (0,1)$, see ...
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2answers
53 views

A stick is broken into two pieces, at a uniformly random chosen break point. Find the CDF.

I'm having trouble understanding how the CDF is found in the solution below: We can assume the units are chosen so that the stick has length $1$. Let $L$ be the length of the longer piece, and let ...
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0answers
12 views

Formula to evenly distribute elements without knowledge of the other buckets

I am trying to write a formula to determine how to evenly distribute elements into individual buckets without specific knowledge of each bucket. The only knowledge that you have is the max number of ...
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0answers
26 views

PMF for sum of uniformly distributed random variables

Let $X_1$ and $X_2$ be independent integer valued random variables that both are uniformly distributed on {1, 2, . . . n}. What is the PMF for S := $X_1$ + $X_2$? What I have so far: P(S=$X_1$+$X_2$) ...
0
votes
1answer
31 views

Product of exponentially distributed and uniformly distributed random variables [closed]

Let $X$ be an exponentially distributed random variable, and let $V$ be a uniformly distributed random variable on $\{-1,+1\}$ that is independent from $X$. Furthermore, let $Y = X \cdot V$. I want ...
0
votes
2answers
30 views

Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
0
votes
2answers
36 views

Computing the probability of waiting someone - Uniform distribution

I have the following problem and I having trouble in finding it solution. I need a hint. The problem: Two people arranged to meet between 12:00 and 13:00. The arriving time of each one is i.i.d. and ...
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0answers
36 views

Statistics $X_{(1)}$ complete for a Uniform Distribution?

Someone had asked this earlier, but since it was good practice for my qualifying exam coming up, I figured I would ask and share my work on the problem. The problem is: Suppose $X$ is Unif$(0,\...
1
vote
2answers
55 views

How do I find the cdf of $X_1 + X_2$?

$X_1$ uniform $(0,1)$ and $X_2$ uniform $(0,2)$ $$ \begin{cases} f(x_1,x_2) = \frac{1}{2}, &\quad \mbox{for} \ 0<x_1<1, 0<x_2<2 \\ 0, & \quad \mbox{otherwise} \end{cases} $$ ...