# 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.

405 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
284 views

410 views

62 views

### Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$Z = -X + W\\ Y = X + XV.$$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
323 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. $\text{Uniform}(0,1)$ 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 ...
322 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 ...
120 views

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 $... 0answers 817 views ### Expected Value - Uniform distribution over infinite interval Question: The probability that an error is introduced into a packet is$\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of ... 0answers 56 views ### What is the difference between these two questions? Consider two independent random variables: X~U(0,1) and Y~U(0,2). Let Z = min(X,Y). b) Find F$_Z$(z) in terms of F$_X$(.) and F$_Y$(.). c) Eliminate F$_X$(.) and F$_Y$(.) to find F$_Z$(z). What is ... 1answer 4k views ### Derivation of Variance of Discrete Uniform Distribution over custom interval I'm trying to prove that the variance of a discrete uniform distribution is equal to$\cfrac{(b-a+1)^2-1}{12}$. I've looked at other proofs, and it makes sense to me that in the case where the ... 2answers 1k views ### Pdf for distance between two uniform random points in a circle This is my first post in the group and I would be very thankful for any help. I am trying to develop a probability distribution for a performance analysis in my thesis. I am trying to look in to ... 2answers 153 views ### Uniform Random Variable on$[0,1]$and Bernoulli$(1/2)$Let$X_1,X_2,...$be independent, identically distributed (iid) random variables with distribution Bernoulli$(1/2)$. Define the random variable: $$Y=\sum_{n=1}^\infty\frac{X_n}{2^n}.$$ Then$Y$is ... 1answer 21 views ### showing the pdf of n-th order statistics I am working on a mathematical stats assignment and I got stuck here. Letting$X_1, X_2, ... ,X_n$a random sample from uniform(0,$\theta$), and Y is n-th order statistic, I need to show that the ... 1answer 44 views ### How to compute a conditional expectation I want to compute a conditionnal expectation, i know that$Z=(Z_1,\ldots,Z_p)'$where$ Z_j=\Phi ^{-1}(U_j)$with$Z \sim N(0,R(\theta))$and$R(\theta)$the$p \times p$positive definite ... 0answers 78 views ### Variance of sum of two uniform RV Let$X$and$Y$be two independent random variables, each uniformly distributed on$[-1,1],$then find$\operatorname{Var}(X+Y).$My attempt : $$\operatorname{Var}(X+Y) =\operatorname{Var}(X) + \... 0answers 98 views ### Show that P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n The lifetime X (in days) of a device has an exponential distribution with parameter \lambda. Moreover, the fraction of time which the device is used each day has a uniform distribution over the ... 0answers 80 views ### “Fragmentation” of a distribution (from paper) I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to ... 0answers 60 views ### Normal approximation of sum of uniform independent RVs using CLT Let X_1, X_2, ... X_{16} and Y_1, Y_2, ... Y_{16} be independent uniform random variables over the interval [-1,1] and let:$$ W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16} ... 2answers 38 views ### On the probability of forming a triangle, when$(0,1)$is divided into three segments , where dividing points are i.i.d. Uniform$(0,1)$Divide$(0,1)$into three line segments, let$X,Y$be the dividing points. Assume$X,Y$are independent and follows Uniform$(0,1)$. What is the probability that the three line segments can form a ... 0answers 208 views ### How can I calculate the joint probability for three variable? I am a student studying the joint probability density function with multi variables. I understand how to obtain a joint probability density function when two uniform distributions have the following ... 1answer 120 views ### calculate marginal PDF from joint PDF of dependent random variables The marginal PDF$f_X(x)$can be calculated as $$f_X(x)=\int f_{X,Y}(x,y)dy=\int f_{X|Y}(x|y)f_Y(y)dy \tag{1}$$ However, I stuck in a particular case as follows.$\mathbf{X}=[X_1,X_2]$is uniform ... 1answer 24 views ### Independent variables' probability functions and expectations So, I've got this exercise Two independent random variables$X,Y$~ Uniform$[0, 1]$. Find the probability function of the random variable$Z=X−Y$. Compute expectation of$E [Z]$So, I need to find ... 2answers 28 views ### Probability in uniform I need help to solve the following problems. Thank you in advance. Problem 1: A random variable$X$is uniform$[0, 1]$. Find the probability that X's 2nd digit is$3$. As far as I understand it ... 0answers 69 views ### Probability of being closest to each other Suppose there are$N$red dots and$M$blue dots uniformly distributed in a square region with side$S$. Each red dot finds its closest blue dot and each blue dot finds its closest red dot. What is ... 0answers 74 views ### Conditional distribution of$Y$given$X+Y$when$X \sim$Unif$(-a,a)$and$Y \sim F$THE PROBLEM HEURISTIC SOLUTION Since$X \sim$Unif$(-a,a)$, the restriction that$X+Y=u_0$automatically imposes the following restriction on$Y$: $$Y \in (u_0-a,u_0+a) \tag{1}$$ Hence the ... 0answers 126 views ### Convergence of ratio of two sums of uniform random variables Consider the sequence of rectangles, which sides are length$(X_1, Y_1), (X_2, Y_2),...,$where$X_1, X_2,...$have uniform distribution on$(0,2)$and$Y_1, Y_2, ...$have uniform distributions on$(...
Sorry if my title makes confusion. Let $\mathbf{x} \in \mathbb{R}^n$ is uniformly distributed on a $(n-1)$-sphere of radius $\sqrt{nP}$, thus $\left\Vert \mathbf{x} \right\Vert^2=nP$. Obviously, \...