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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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27 views

Interpretation of sentence to pdf

Just a quick question on interpreting what this pdf looks like: Bacteria are distributed randomly and uniformly throughout river water at the rate of $\lambda$ bacteria per unit volume. n test tubes ...
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1answer
134 views

Expected value of the shifted inverse of a binomial random variable, and application

Here is an exercise given by a colleague to a student : Let $X\hookrightarrow B(n,p)$ and $Y=\frac{1}{X+1}$. Find ${\rm E}(Y)$. It is not very difficult to prove that the answer is $${\rm E}(Y) = ...
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24 views

A doubt in finding the expected value of lifetime

Lifetime of a bulb has uniform probability distribution on (2,12). Bulb is replaced upon failure or upon reaching age 10, whichever occurs first.Find the expected value and standard deviation of age ...
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80 views

Variance of sum of two uniform RV

Let $X$ and $Y$ be two independent random variables, each uniformly distributed on $[-1,1],$ then find $\operatorname{Var}(X+Y).$ My attempt : $$\operatorname{Var}(X+Y) =\operatorname{Var}(X) + \...
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1answer
45 views

uniform distribution with interval (0,2) and sample 12

Q): Suppose that you wish to sample $12$ observations randomly from a uniform distribution on the interval $(0,2)$. An approximate value of the probability that the average of your sample will be less ...
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1answer
36 views

Given a function that generates random numbers with uniform distribution over (0, 1) find a function to generate numbers with Bernoulli distribution.

If we have a continuous random variable $X$ with uniform distribution over $(1, 0)$ we can find functions that generate numbers with other distributions using this random variable. For example if we ...
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71 views

Probability that Alice and Bob keep dating infinitely often

I solved the following problem. I would appreciate it if you can please provide feedback and let me know if I have made any mistakes. Problem statement: Online dating: On a certain day, Alice ...
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1answer
31 views

P(X>Y) when X and Y are continuous uniform distribution

Suppose $X$ and $Y$ are continuous uniform random variables. If $X \sim U[a,b]$, $Y \sim U[c,d]$ and $[c,d] \subset [a,b]$ find the probability that a random $X$ value is greater than a random $Y$ ...
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1answer
338 views

$E(X^2)$ of discrete uniform distribution

I have a discrete uniform distribution (from IFoA formulae) with parameters a, b, h, where a and b are the start/end points, and h is the interval between each value. The p.d.f. is given as $\frac{h}{...
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43 views

PDF of the product of two independent uniformly distributed random variables

Suppose that X and Y are independent U[0,1]-random variables. Find the probability density function of the product V = XY. I have seen that 𝑓(𝑧)=(−1)^(𝑛−1)log(𝑛−1)(𝑧)/(𝑛−1)! for the product of ...
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41 views

Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
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Why is quantile function of uniformly distributed random variable a random variable?

I have the quantilfe function $F^{-1}$ of a random variable which is defined as: $F^{-1}: ]0, 1[ \ni u \rightarrow F^{-1}(u) = inf\{x: F(x) \geq u\} \in \mathbb{R}$. Now I can define a new random ...
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1answer
90 views

Relationship between cdf of normal distribution and uniform distribution defined on $[0,1]$

So the problem is, given $X\sim N(1,2^2)$, $Y=e^X$, if the cdf of standard normal distribution is $\Phi$, to show that $$\Phi\left(\frac{\ln Y-1}{2}\right)\sim U[0,1],$$ where $U[0,1]$ is the ...
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50 views

Citation for marginal of uniform hypersphere distribution

Based on some other answers (such as Marginal Distribution of Uniform Vector on Sphere), I understand that the marginal of a uniform distribution on a hypersphere is a Beta distribution. Is there a ...
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1answer
91 views

Computing $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$ where $X_1,X_2,X_3$ are i.i.d $U(0,1)$

Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_{(1)} , X_{(2)} , X _{(3)} )$ be the corresponding order statistic. Compute $\mathbb{P}\{X_{(1)}+X_{(...
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Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in [-\alpha,\alpha]$, $Y\in [-\alpha,\alpha]$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
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64 views

Compute $P(X_{(2)} ≤ 3X_{(1)})$ by using the integration technique

Suppose that $X_1,X_2,X_3.X_4$ are independent $U\in(0,1)$-distributed random variables and let $(X_{(1)}X_{(2)}X_{(3)}X_{(4)})$ be the corresponding order statistic. Compute $P(X_{(2)} ≤ 3X_{(1)})$ ...
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Conditional expectation of uniform random variable given order statistics [closed]

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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266 views

How to Find the CDF and PDF of Uniform Distribution from Random Variable

If a random variable is defined as $Y = 3 - 2X$, how do I find the CDF and PDF of $Y$ if $X$ follows a Uniform distribution of $X \sim (-1,1)$? For CDF, am I trying to solve for $F(b) = F(-2)$ or $F(...
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Bivariate density with uniform marginals

I kindly ask for your help to solve this problem. Consider two standard uniform random variables $X_1,X_2\sim U[0,1]^2$. Then, the questions are: 1) is it possible to find the explicit form of joint ...
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1answer
200 views

Meaning of “uniformly sampling”?

Reading an article I came across the following expression: '' $\overline{X}$ is constructed by sampling uniformly along the straight line between the pair of $X$ and $\hat{X}$." I know what a ...
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1answer
37 views

Probability of nth event is between x and y

I have a uniform distribution of age in the range $[a, b)$ with $a=42$ and $b=78$ So the probability that a person walks in a bank that is between $50$ and $70$ years of age would be $\frac{70-50}{78-...
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287 views

Hypothesis Testing under Uniform Distribution Question

The question reads: Let $\theta > 0$ and $X \sim \mathcal{U}[0, \theta]$, i.e. $X$ is uniformly distributed on the interval $[0, \theta]$. Assume that $\theta$ is unknown, but we can observe $X$. ...
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133 views

Uniform distribution variable in the Newton symbol

The following question is from actuarial exam. Let $N$ be uniformly distributed on $\{0,1,2,...,19\}$. Compute $$\mathbb{E}\sum_{k=0}^{N}{N-k \choose k}(-1)^k$$ I started $$\mathbb{E}\sum_{k=0}^{N}{N-...
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11 views

Expected number of neighbors in an Area with Random Uniform distribution

In an area of ${100m^2}$ 19 sensor nodes are uniformly and randomly distributed. Root node is located at the center of this area. Each node has a transmission range of 40 meters in all directions. All ...
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21 views

MLE for special uniform case [duplicate]

I'm preparing for my exam and I came across this question: Let $X_1, X_2, ...,X_n$ be iid with PDF $f(x)=\frac{2x}{\theta^2}$ for $0 \leq x \leq \theta.$ Find the MLE of $\theta$ So this is what ...
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1answer
49 views

If $X_1$ and $X_2$ are uniformly distributed random variables with parameters $0$ and $1$, what is the distribution of $Y = X_1 + X_2$?

I was doing a recap of the probability theory I had last year and even though this question shouldn't be hard, it is somehow confusing me immensly. Clearly, if we have $X_1, X_2$ belonging to $U[0, ...
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114 views

Method of Moments estimators of $\alpha$ and $\beta$

Let 5 numbers 2, 3, 5, 9 and 10 come from a uniform distribution on the interval $[\alpha,\beta]$. Find the method of moments estimators of $\alpha$ and $\beta$. Any help would be appreciated, thank ...
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1answer
135 views

MLE of $\theta$ in $U[0,\theta]$ distribution where the parameter $\theta$ is discrete

Consider i.i.d random variables $X_1,X_2,\ldots,X_n$ having the $U[0,\theta]$ distribution: $$f_{\theta}(x)=\frac{\mathbf1_{[0,\theta]}(x)}{\theta}$$ , where the unknown parameter $\theta\in\{1,2,\...
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1answer
22 views

Would this method approximate a uniformly distributed random points on sphere?

I know there are several ways to generate uniformly distributed random points on the 2-sphere $S^2$. But I would like to know if my method does the same job, although it is very inefficient. Say, I ...
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32 views

What is the name of this principle?

When generating uniformly-distributed samples from a multidimensional distribution, I believe that sampling each dimension independently produces uniformly-distributed samples from the original ...
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1answer
2k 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 = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \...
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1answer
135 views

Use the change of variables to determine the density for a uniform distribution on $[a,b]$

Knowing that the density of a uniform random variable on $[0,1]$ is: $f_{U}=\left\{\begin{matrix} 1 & x\in [0,1]\\ 0 & x\notin[0,1] \end{matrix}\right.$ How to determine the density of a ...
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2answers
160 views

Pdf of $X+Y+Z$ where $X,Y,Z$ are independent $U(0,1)$

This is my working out of the problem so far, I want to know if there is a more simpler way to solve this, or I would just be interested in other methods that one could use to solve a similar problem
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25 views

PDF of conditional uniformly distributed random variable

Given two barrels of water, $A,B$, with 1 liter each. We pour an $X\sim U[0,1]$ amount of water from $A$ to $B$ and then $Y$ amount of water randomly from $B$ to $A$ $(Y|X=x\sim U[0,1+x])$. ...
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23 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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11 views

Average Number of users based on some condition and error calculation

I want to Calculate average No. of users selecting any particular No from random number range (2 to 6) and I have defined error condition as when more than one user selects same number. Now based on ...
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3answers
78 views

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases?

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases? I first found the density functions: $$ f_x(t) =\begin{cases} 1 &...
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1answer
38 views

The probability of sum $x+y$ to be greater than $20$

The variable $x$ takes a value between $0$ and $10$ with uniform probability distribution.The variable $y$ takes a value between $0$ and $20$ with uniform probability distribution. The probability of ...
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2answers
29 views

Distributing 6 persons in their shifts evenly with rotating partner

I thought this was easy but I was mistaken. I need to make a schedule for 6 persons. They have to make shift by two (partner) but the partner should also rotate so that all person gets to be ...
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1answer
32 views

Self-study Order Statistics

So I got this exercise from a book and I'm confused by a statement they made. Example: In a 100-meter Olympic race, the running times can be considered to be $U$~$(9.6, 10.0)$-distributed. Suppose ...
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292 views

Is modular multiplication under a prime modulus uniformly distributed?

Given a prime $p$ and $m \in Z_p^*$. Assume we draw $a \stackrel{u}{\in} Z_p^*$ uniformly at random. Will $a \cdot m \; mod \; p$ be distributed uniformly over $Z_p^*$?
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172 views

Doubt in proof for Complete Statistic for Uniform Distribution

A statistic $T(X)$ is called complete statistic for a parameter $\theta$, if $E_{\theta}g(T) = 0$ for all $\theta$ implies $P_{\theta}(g(T) = 0) = 1$ for all $\theta$. I interpret $P_{\theta}(g(T)...
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2answers
55 views

Uniform Distribution - Is my solution correct?

I'm preparing for an exam in Probability and Statistics and since I am self-studying, I don't have answers to refer to. So would be really great if someone would please take the time and check if my ...
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1answer
64 views

$U_1,U_2,…$ i.i.d. $U[0,1]$, $P\sim \mathrm{Poi}(\lambda)$, find $F_{\operatorname{min}(U_1,…,U_P)}$

Let $(U_n)_n$ a sequence of random variables i.i.d $U[0,1]$ and let $P\sim \mathrm{Poi}(\lambda)$ a random variable such that $P$ is independent of $(U_n)_n$. Let $$ \\ X=\left\{\begin{matrix} \...
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1answer
78 views

Uniform Distribution: Expectation $E(\bar{X})$

Uniform Distribution: Expectation $E(\bar{X})$ Assuming that the samples are independently and identically distributed (iid): First i was asked to find $E(X_k)$ $E(X_k)= $$\int_{0}^{θ} dn P_θ(n) n =...
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1answer
40 views

Generating a number belonging to N(0,1) using *m* numbers from U(0,1) using central limit theorem

I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ. As a starting point, author uses generated uniform ...
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0answers
98 views

Show that $P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n $

The lifetime $X$ (in days) of a device has an exponential distribution with parameter $\lambda.$ Moreover, the fraction of time which the device is used each day has a uniform distribution over the ...
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2answers
76 views

find the probability about sum of random variables

Let $X_1, X_2, X_3, Y_1, Y_2, Y_3, Z_1, Z_2, Z_3$ be random variables which have uniform distribution between 0 and 1. It means, the average of $X_1 = 0.5$ Let: $X=X_1 + X_2 + X_3,$ $Y=Y_1 + Y_2 + ...
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1answer
1k views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...