Questions tagged [uniform-distribution]

The uniform distribution describes a random variable that is equally likely to take any value in its sample space.

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Finding probability involving dependent random variables [closed]

Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between ...
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Distribution IID uniform variables given their ranking [duplicate]

Description Let $N\in\mathbb{N}^{+}$ and $X_{n}\stackrel{IID}{\sim}U(0,1)$ for $n\in\{1,...,N\}$. Given $X_{1}\leq X_{2}\leq X_{3}\leq...\leq X_{N}$, I would like to understand $f_{X_{n}}$ by writing ...
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Complete statistic for joint distribution of independent variables [closed]

If $X_1, ..., X_m\sim U(0,\theta)$ and $Y_1, ..., Y_n\sim U(0,\theta ')$, then $X_{(m)}$ and $Y_{(n)}$ (the last order statistics) are complete sufficient for $X$ and $Y$. Is there a straightforward ...
Charlie Roth's user avatar
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Method of Moments of Uniform Distribution

Let $x_1=2, x_2 = 1, x_3 = \sqrt5, x_4 = \sqrt2$ be the observed values of a random sample of size 4 from a uniform distribution $U(-\theta, \theta)$ where $\theta>0$. Then the method of moments ...
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Sampling 1 item from groups of correlated values and combining the statistics

We have a dataset consisting of several groups of observations: Group Object Value Gr_1 Ob_1 V_1 Gr_1 Ob_2 V_2 Gr_2 Ob_3 V_3 ... ... ... All values lie in the interval [0,1]. In each of the ...
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Discrete test statistics cannot form a uniform P-value?

I received this feedback on my permutation test design from a collaborator and I'm wondering if his claim is valid. My test statistics are discrete (like counting the number of red marbles found after ...
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Joint density of two functions of a uniformly distributed random variable

I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$. I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
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What is this hybrid(mixed) random variable’s variance?

X ∼ Uniform(a,b), a<b (Discrete) where f(x)=1/n where n=b-a+1 and Y ∼ Uniform(c,d), c<d (Continuous) where g(y)=1/d-c. X and Y are independent. Let z = x - y. I was able to find the E(Z), ...
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Recursive Uniform Distribution Expectation Question

Suppose we draw some k ~ Unif(0, 1). Then, we will draw some $u_1$ ~ Unif(0, 1). If $u_1 < k,$ we stop. Else, we will draw $u_2$ ~ Unif(0, $u_1$). We will continue drawing until $u_n < k,$ where ...
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Random correlation matrices

Suppose that we simulate random $n\times n$ correlation matrices by assigning iid $U(-1,1)$ random variables to all off-diagonal entries and accept matrices $\boldsymbol\Sigma$ that are positive ...
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transformation of uniform random variables

Let $U_1, U_2,...,U_n$ be a sequence of independent random variables with Uniform distribution over the interval $(0, 1)$ and let $Y = -\frac{1}{\lambda} log(U_1)$ . what is the distribution of Y? i ...
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Weighted Average of Uniformly Distributed RV [duplicate]

Let $x \sim U[0,1]$ and $y\sim U[0,1]$. Let $z= \omega\, x+ (1-\omega)\,y$, where $\omega\in[0,1]$. The pdf of $z$ is a trapezoidal distribution over $[0,1]$: \begin{equation*} \begin{aligned} f(z)&...
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On the estimated formula of covariance of two random variables

We define the covariance of two random variables $X$ and $Y$ as $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$. The covariance measures the "linear dependence" between the two r.v s. But in a lot of ...
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Uniformly sampling surface of an ellipsoid using multivariate Gaussian

Sampling uniformly from the surface of an ellipsoid (in the sense of $\mu(dA) = \frac{1}{A}$) seems very nontrivial: How to sample uniformly from the surface of a hyper-ellipsoid (constant ...
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Distribution of positive semidefinite matrices that are generated by uniformly distributed positive definite matrices

Let $\mathcal{A}=\{ A_1,A_2,\dots,A_n \} \subseteq \mathcal{S}^p_{++}$ be a set of real positive-definite matrices sampled uniformly with a fixed trace (say, using this algorithm). To convert each $...
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Conditional Expectation in Uniform Case

Let $X$ and $Y$ be independent random variables where $X \sim uniform[\underline{x}, \bar x]$ and $Y \sim uniform[\underline{y}, \bar y]$. What is the conditional expectation of $X$ given $z = X + Y$? ...
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Distribution of a sum of linear combinations of random variables, each drawn from a set of random variables

Question. Let $X_1, X_2, ..., X_n$ be a set of normal random variables, each with variance ${\sigma }^2$ and mean 0. For each $i,j$ in pair in $X$, $Cov(X_i,X_j)=V$. Further, let $Y_1, Y_2, ..., Y_m$ ...
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Distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d $\text{Exp}(1)$

Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$. Simulation suggests the ...
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How to interpret a qq plot of uniform distribution whose slope is greater than 1

I am trying to interpret a qq plot of a uniform distribution in R where the plot is as shown in the image. The qq lines are a kind of straight but the slope of these lines way greater than the 45 ...
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Probability distribution derivation given histogram of outputs

I'm not too versed in statistics, but I am currently dealing with a problem that pertains to probability. If any assumptions are off on my part, please correct me. I have a 2D polynomial function of ...
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What is the distribution of a uniform with a bound drawn from a uniform?

Suppose I have a uniform distribution $X \sim U[a,1]$ with $a \sim U[c,1]$? How can I characterize the CDF of X?
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Why is the distribution of the sum of the values on two dice bell-shaped and symmetric if two uniform dist is triangular distribution?

Why is the distribution of the sum of the values on two dice bell-shaped and symmetric if two uniform dist. sum is triangular distribution via Irwin-hall distribution?
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Calculate the long-term probability

The following question is an interview question about probability: There is a list of items and how many times each item is purchased (range from 10 to 100,000 times). The probability of users buying ...
Anne Maier's user avatar
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Probability Density of the Sum of Two Un-identical Uniform Random Variables

Let $X$ ~ Uniform$[a,b]$ and $Y$ ~ Uniform$[c,d],$ where $a\le b\le c\le d.$ Find the probability density of $Z = X + Y.$ I know I have to use the convolution formula $$f_Z(z) = \int_{-\infty}^\...
roosters0405's user avatar
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Power of Uniform Order Statistics

I know that if $U$ is a uniform r.v. in $(0,1)$, then $U^a\sim Beta(1/a,1)$ with $a>0$. On the other hand, if $U_{(1)}\leq \cdots\leq U_{(n)}$ are the uniform order statistics, then, with $U_{(0)}=...
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Two dimensional random variable with uniform marginal probability density functions [duplicate]

I have access to some data for two variables - let's call them x and y. In particular, I have the distribution of data separately per each variable, something that allows me to estimate the marginal ...
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Are two marginal distributions of a student-t copula equivalent to using two independent uniform distributions?

I am trying to figure out if these two are the same: Using the marginal uniform distributions of a student-t copula Using independent uniform distributions I have generated SAS code to figure this ...
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Uniform distribution in the target variable

I'm analysing a dataset that gave some very poor regression models in the past (not trained by me). Basically, I'm trying to get insights on why those models were so bad. The target variable is the ...
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Uniform Sampling From the Region Bounded by $\sqrt{x}$, $x=3$, and $y=0$

I want to sample uniformly from the area bounded by $y=0$, $x=3$, and $y=\sqrt{x}$: If I draw $x$ from $U[0, 3]$ and $y$ from $U[0, \sqrt{x}]$, the density will be higher in the bottom left corner: ...
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Am I using the chi-squared test correctly?

I have a set of $2,142$ measurements of some value that are grouped into $18$ bins of equal length (according to the value measured). I want to check the resulting distribution for uniformity. As far ...
Eugene B.'s user avatar
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Regression with random vs uniform distribution of a covariate

Suppose we have some population of individuals, each having X and Y variable. I know that in the population, most individuals have low values of X, but I want to find regression formula for low as ...
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How do I create random teams of people, each person has multiple parameters, and the parameters are equally distributed among teams

As the title suggests, I am trying to divide a group of people randomly into different teams of equal size. However, each person has parameters that go with them (such as age). If there is only one ...
Stephen's user avatar
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Asymptotic distribution of $n^r \frac{U_{(1)}}{U_{(n)}}$: figuring possible $r>0$

Consider the i.i.d. sample $U_1, U_2 \cdots, U_n$ from the uniform distribution $U(0, 1)$. I should find a possible values of $r>0$ to have an asymptotic distribution of $$ n^r \frac{U_{(1)}}{U_{(n)...
ToBY's user avatar
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Conditional probability density of the ratio of two independent uniform random variables with different supports

Let $X = B * [(u + \epsilon_u) - C]$. $u$ represents a true measurement value. $\epsilon_u \sim U(-0.5, 0.5)$ represents the error associated with that measurement value. $u + \epsilon_u > 0$. $B$ ...
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Conditional probability density of the sum of an uniform random variable with a constant

I am interested in the conditional distribution of the sum of a uniform random variable and a constant. Let $X = d + \epsilon_d$. $d$ is the true measurement value. $\epsilon_d$ is the error in the ...
BeginnersMindTruly's user avatar
4 votes
3 answers
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Is it possible to uniformly draw points over a $D-2$ sphere, given that one has an algorithm to draw over the $D-1$ sphere in D-dimensional space?

Suppose I have the following scenario: And I am aware of an algorithm to draw uniformly from (in this case) the 2-sphere. Does this same algorithm readily extend to the situation where I randomly ...
tisPrimeTime's user avatar
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Distribution of the product of an inverse uniform distribution and a constant

Let's assume I have a measurement $d + \epsilon_d$ where the true value of the measurement is $d$ and the error associated with that measurement $\epsilon_d$ has a uniform distribution $\epsilon_d \...
BeginnersMindTruly's user avatar
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Uniform distribution and Gaussian distribution [duplicate]

As the number of values sampled from the uniform distribution increases, the distribution of the mean tend towards a Gaussian. Why is that?
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Uniqueness of a Latent Representation Under Monotonicity Condition?

Suppose that I observe a bi-variate joint distribution over two random variables, $(X_1,X_2)$. I want to represent this joint distribution as arising from a function $F$ applied to i.i.d. uniform ...
stats_model's user avatar
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Drawing samples from a joint distribution defined by limits?

Assume that I want to efficiently draw samples from a (for simplicity bivariate) joint distribution $p(x,y)$, with $x \in \mathbb{R}$ and $y \in \mathbb{R}$. I don't have a closed-form expression for $...
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test for deviation from uniform distribution [closed]

Imagine I have a 3x3 matrix of the type: [0.22 0.15 0.99 0.28 0.42 0.51 0.16 0.76 0.12] The numers in the matrix are the result of a complex model, please ...
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Variations on Uniform Priors with Binomial Signals? [duplicate]

the standard result is that with a uniform prior on p (from 0 to 1) and binomial signals (h H signals and (n-h) L signals from n draws, each with probability p), the posterior mean is (h+1)/(n+2) and ...
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Generating uniformly distributed particles on a $n$-dimensional flat torus or periodic hypercube [closed]

I am trying to generate evenly distributed particles in an $n$-dimensional flat torus or a periodic hypercube. I am not sure if any of this approaches suffices. Can you suggest alternative methods for ...
Rober's user avatar
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Does conditional probability that equals zero imply the events are disjoint

Does $P(B|A) = 0$ with $P(A) \neq 0$ mean $A \cap B = \varnothing$? I think I already have an answer, but I'm not sure it's correct. I would say no, because we can consider a variable $X \sim U(0,1)$, ...
Valikeny's user avatar
4 votes
2 answers
213 views

Transform bivariate uniform variable

Let $X_1 = U(0,1)$ and $X_2 = U(0,1)$. $X_1$ and $X_2$ are independent. Then $f(x_{1}, x_{2})=1, {0}\le{x_1}\le{1}, {0}\le{x_2}\le{1}$. Let $Y_1 = \arctan(X_{2}/X_{1})$, $Y_2 = X_2$. I need to find ...
Igor Yegin's user avatar
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Definition p-value and find p-value in practice

I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $...
Samvel Safaryan's user avatar
7 votes
1 answer
213 views

Donut-like Distribution in Cartesian Coordinates

I have a set of points $P_i$ which are described by an angle $\theta_i$ and a magnitude $r_i$. $\theta_i$ follows a Uniform distribution $(\theta_i \sim U(0, 2\pi))$ and $r_i$ follows a chi-k ...
Liam F-A's user avatar
1 vote
0 answers
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Test for uniform distribution with multiple participants

I know how to use e.g. the Chi-Square test to check whether data are uniformly distributed. I was wondering whether the approach changes if the data comes from multiple participants, i.e., there is a ...
einGlasRotwein's user avatar
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UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta) $ where $\theta \in[1, \infty)=\Theta$

Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
Debarghya Jana's user avatar
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Testing whether a set of points on the unit sphere is uniformly distributed

The canonical way to do the test is to perform the spherical harmonic transform of the empirical distribution and then check that the power spectrum decays, but this is presumably fairly expensive. Is ...
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