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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KL-Divergence on Uniform Distribution, is this correct?

$P_1$ is uniform on $[0, 0.5], ~P_2$ is uniform on $ [0, 1] $ What is the KL-divergence $(P_1 \Vert P_2)?$ Attempt: $$ D(p\Vert q) = \sum_{x\in X} p(x) \log \frac{p(x) }{q(x) }.$$ KL$(P_1\Vert P_2) =$...
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Are there better measures of entropy

Related question here I am trying to measure the uniformity of multimodal distributions and am looking into using entropy. I would like a measure of entropy that is higher for the first distribution ...
8 votes
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Why does somebody argue that the number of bootstrap replications should not be a multiple of 10?

At a recent conference somebody claimed that the size of the bootstrap replications should always be 999 rather than 1000. Which argument supports this claim?
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How to Justify this Two-Sided Test is UMP with NP Lemma?

UMP tests generally do not exist for two sided tests, ie $H_0: \theta = \theta_0$ vs $H_a: \theta \neq \theta_0$. However, if we observe $n$ iid observations of $X\sim Unif(0,\theta)$, we can ...
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Joint Uniform Distribution Probability Problem

Let $X \sim U(0,1) $ and $Y \sim U(0,x) $. Calculate $$ \Pr(X >0.5 | Y= 0.25)$$ Is this a trick question ? Since $\Pr(Y = 0.25) = 0$, right ?
7 votes
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General form of the distribution of distances to a fixed point in rectangle

The answer given to the question Probability distribution of the distance of a point in a square to a fixed point solves for the "distribution of the distance between the origin (0,0) and a ...
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Convergence of a function having a big summation at each sample

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ Here, $U_m$ samples are random numbers coming from a Gaussian distribution $$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$ and ...
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2 answers
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Probability random variable is less or equal to k-th out of two samples when ordered

Given the random variable $X$, $\{X_{i}\}_{i=2}^{n}$, $\{Y_{i}\}_{i=2}^{n}$ all iid and lets denote $X_{(k)}$ as the k-th statistic of $\{X\} \cup \{X_{i}\}_{i=2}^{n}$ and $Y_{(k)}$ for $\{X\} \cup \{...
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QR interview problem Guessing order of draws from iid U(0,1)

This is for QR at two well know trading firms (think jane street, HRT, Citadel, Jump ...)(not BB bank). Question prompt: Given n iid Uniform distributed r.v.s. $x_i$ ~ U(0,1). $x_1$ is drawn first, ...
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Extracting statistical parameters from a mixture of two distributions of different kind

I have a dataset b (as a list in Python) of length 100 I know that is amounts to the mixture of two distributions: A normal distribution A uniform distribution ...
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Does box muller algorithm produce independent and identically distributed samples? [duplicate]

Since, box muller generates two samples of standard normal distribution can we say that it produces IID samples
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Finding the MVUE of the center of a circle of unknown location

Is there a known analytic solution for finding the minimum variance unbiased estimator of a disk of an unknown location given that a sample of $n$ points was drawn uniformly and randomly from the disk ...
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Conditional distances in order statistics

Assume I have $n$ points sampled independently from the uniform distribution on the unit interval. After ordering the sample I get the points $X_1, X_2, \dots X_n$ such that $X_1 \leq X_2 \leq \dots \...
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probability that the players will exchange their initially drawn number

Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own ...
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Find the MLE density function of uniform [-\theta,\theta] [duplicate]

For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-...
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1 answer
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MLE of the Uniform Distribution

In a uniform distribution where $0\leq X \leq \theta$, the pdf is represented as $f(X|\theta) = \frac{1}{\theta}I(0\leq X \leq \theta)$, and the likelihood is $L(\theta) = \prod\frac{1}{\theta}I(0\leq ...
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MLE for the Uniform distribution [duplicate]

I understand how a random sample $x_1, ..., x_n$ following the Uniform Distribution with $0 \leq x \leq \theta$ has a log-likelihood proportional to $\frac{1}{\theta^2}$. I am told that the MLE for $\...
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What distribution to sample X from to get an uniform distribution in Y?

I have a random variable $X$ which is related to another random variable $Y$ as $Y = \text{cos}(X)$, where $X \in [0, \pi/2]$, and I would like to know what distribution I should sample $X$ from in ...
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Find a maximum likelihood estimator in discrete uniform distribution

$X$ is distributed as discrete uniform distribution on $\{1,2,3,...C\}$ where $C$ is an integer higher than or equal to 4 $Pr(X=x)=p$ for $x ∈ \{1,2,\dotsc,𝐶\}$ which means $Cp=1$ $X$ is censored and ...
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Test uniformity after adding some non-uniform data in python

I generate N1 numbers from uniform distribution using numpy python package from a certain interval e.g., ...
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Sample uniformly from unit square conditioned on sum and product

Consider the following conditional distributions: \begin{align} X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X + Y = a & a \in [0, 2] \\ X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X ...
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Uniform measurement error in "errors in predictors" regression

I'm working with cancer incidence data that uses a range of ages (e.g. <1, 1-4, 5-10, ...) rather than a single value. I want to fit a model where age is a predictor. As a result, I'm curious ...
3 votes
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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How to sample and compute the likelihood from a Mollified Uniform distribution?

I want to draw samples from the mollified Uniform distribution presented in another Cross Validated thread, cf the answer from whuber. What is the best way to do so?...
2 votes
1 answer
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Probability that a Random Variable is Greater Than Another

Say I have two independent random variables - one is drawn from a uniform distribution on [0, 50] while the other is drawn from a uniform distribution on [0, 100]. How would I calculate the ...
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Sampling Puzzle

There is a bag with N = 50 balls. Among which M = 10 balls are red, and N-M = 40 balls are blue. Further, say the red balls are numbered among themselves from 1 to 10, and the blue balls are numbered ...
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Probability of failure of Uniform Sampling [duplicate]

Say I have a bag with 10 numbered balls, and I pick one ball at each time step and then put it back in the bag. Since each ball is equally likely, therefore the current situation represents a uniform ...
3 votes
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Is the parameter of interest uniformly distributed over the confidence interval?

Let's assume that we are estimating a parameter of the population (e.g., mean height or weight). We've gathered our sample and calculated confidence intervals around the parameter of interest. Are we, ...
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Confusion over probabilities and densities in a Bayesian Data Analysis textbook question

In Gelman's "Bayesian Data Analysis" question 1.5a), we are asked to estimate the probability of a tie in an election. The question: Probability assignment: the 435 U.S. Congressmembers are ...
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Sampling uniformly using binary representation of a number [duplicate]

I have a $6$-sided dice and I would like to sample integers uniformly from $0$ to $k$ with $k > 6$. I think that $k$ should be written in its binary representation. Let's say its binary ...
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Expected number after n rounds of uniform~[0,1] draws

If we have a series of $n$ IID random variable $X_i$ that are uniform [0,1], and at each round $i$ we decide to either keep $X_i$ or discard it for the next number. What is our strategy to maximize ...
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Expected number of points in a subsurface of a rectangle

There is a rectangle $S$, and $n$ points are uniformly distributed inside it. If we select an area $A$ inside the rectangle, what is the expected number of points inside the $A$? I think it seems to ...
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Distribution of a sample of uniformly distributed points in the 2D

Let there be a rectangle in the plane and a set of points distributed in the rectangle by a uniform distribution. I select a random point on the top and right border and draw the red line. The blue ...
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Are my sequential random numbers correlated?

Say I want to generate $n$ random numbers $Y_i, i = 1, ... n$, where $X_i \in [1,..n]$ is my i.i.d. uniform random source and $Y_i$ are drawn sequentially: $Y_0 = X_0$ $Y_i = (Y_{i-1} + X_i)\; \text{...
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How to show $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $?

Told to show that: if $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $. [where X,Y are continuous random variables] I can see why it holds ...
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Reweight a distribution after draws

I have a practical sampling problem, and I wasn't sure whether there was any off-the-shelf literature to deal with the issue. I am first sampling from a intervals of an approximately equal length from ...
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Estimating $\theta$ based on censored data when $X_i\sim \text{Uniform}(0,\theta)$ with $\theta\ge 1$

Suppose $(X_i)_{1\le i\le n}$ are i.i.d $\text{Uniform}(0,\theta)$ random variables where $\theta \ge 1$. We observe $Y_i=\min(X_i,1)$ instead of $X_i$. I wish to estimate $\theta$ based on the data $(...
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Show that unimodal distribution variance is smaller than uniform distribution

Let $f(x)$ be a unimodal distribution on bounded interval $[-1,1]$. Can we show that the variance is upper bounded by variance of Unif$[-1,1]$?
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How to prove $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$

I've been reading about uniform distributions but I can't see how $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$. Is there a way to prove this using the cumulative distribution ...
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Exponential Posteriori with a Uniform Prior

I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you. Let be $X|\theta\sim U(0,\theta)$ ...
2 votes
1 answer
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Computing KL divergence between uniform and multivariate Gaussian

Another post has addressed the fact that KL divergence is defined between a uniform distribution and a Gaussian distribution $$D_{\text{KL}}(\mathcal{U}(x) \parallel \mathcal{N}(x \mid \mu, \Sigma)) = ...
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1 answer
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Probabilities manipulation $P(Z_1+Z_2=2,Z_1=1,Z_2=1)=P(Z_1=1,Z_2=1)$ with $Z_1,Z_2 \in \lbrace 1,2,3,4 \rbrace$ independent and uniform

I've two independent and uniform random variables $Z_1,Z_2 \in \lbrace 1,2,3,4 \rbrace$. Can you tell me why the following equality is correct please? $$ P(Z_1+Z_2=2,Z_1=1,Z_2=1)=P(Z_1=1,Z_2=1) $$
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Is there any connection between these two distributions?

I was playing with standard uniform distributions where I noticed a "weird" relation between two combinations and was wondering if there was an underlying reason for it (or if it is just ...
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Sum of Discrete Uniforms, but each value can be picked no more than N times?

Suppose there are i.i.d. variables $X_{1,..n}$ with discrete uniform distribution with the support $[1, n]$. What would be the distribution of such a sum if we introduce the condition that each value ...
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$Y$ has uniform distribution on [0,1], and conditional on Y = y, let X have a distribution of Bernoulli(y). What's P(Y|X=1)?

Using Baye's formula I have $P(Y=y|X=1) = \frac{P(X=1|Y=y)*P(Y=y)}{P(X=1)}=\frac{y*1}{P(X=1)}$ Now $P(X=1) = P(X=1|Y=0)P(=0)+P(X=1|Y=0.001)P(Y=0.001)+...P(X=1|Y=1)P(Y=1) = 0+0.001+0.002...+1 = \int_0^...
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In Bayesian models, can you use Uniform(-inf, inf) as a prior?

In Bayesian models, can you use Uniform(-inf, inf) as a prior? I ask because in an class, we looked at MH MCMC sampler, and showed that to sample from a distribution, we need not explicitly solve for ...
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What distribution is produced by the correlation of random values sampled uniformly?

I'm trying to figure out what this distribution is so that I can calculate the exact probability of values close to 1 or -1 using its PDF: as produced by the following code in R: ...
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ks test for uniform distribution

My question is similar to this post (KS test for Uniformity). KS p-valuse changes when samples are in different position, even if they have the same intervals. For example: ks.test(c(1,2,3),"...
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Scipy: Continuous Uniform Distribution's CDF

Question How can we solve the problem below with Scipy? Background: I am preparing statictics notes in Jupyter Notebooks and trying to apply appropriate Scipy functions to some real cases. Problem: I ...
4 votes
2 answers
156 views

Find marginal distribution of Y while knowing distribution of X and $Y|X$

Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given $X = x$ is a binomial distribution with parameters $(n, x)$. Then we say that Y has a binomial ...
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