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

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Find the sampling distribution of the MLE of the uniform distribution [duplicate]

The MLE is $ \theta = max [x1,...,xn] $ And $ P(max [Xi] < t) = P(Xi < t)^n = P(t/\theta) $ But the question asks me to show that $ P(max[Xi]< t) = (min[\theta, t]/ \theta)^n * I[t>0] $ ...
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85 views

Show that $\min(U,1-U)$ and that $\max(U,1-U)$ are uniform

Let $U$ be uniform on $(0,\ 1)$. Show that $\min(U,\ 1-U)$ is uniform on $(0,\ 1/2)$ and that $\max(U,\ 1-U)$ is uniform on $(1/2,\ 1)$. I'm not sure how to approach... the only hint i have is that a ...
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Advantages of Box-muller over inverse CDF method for simulating Normal distribution?

In order to simulate a normal distribution, from a set of uniform variables, there are several techniques: The box muller; in which one samples two independent uniform distributions $(0,1]$ and ...
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Determine the limiting distribution of Uniform Order Statistic

I have a random sample of size $n$ from a uniform distribution $$U(0, \theta)$$ And I've proven that the pdf of $Y_n$, the n-th order statistic of the sample is: $$ f_{Y_n}(y) = \frac{n}{\theta^n} ...
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Maximum of uniformly distributed random variables using iterated expectations

I'm working through the problems in Wasserman's 'All of Statistics'. The chapter on expectations and conditional expectations ends with a (seemingly) easy problem: Let $Y$ be the maximum of $n$ iid ...
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48 views

Remapping the Sum of two Normal Random variables?

I have a problem where I have sum of two random variables 1). Each distributed independently normally with different means ($\mu_1$, $\mu_2$) and sds ($\sigma_1$, $\sigma_2$). $Z=R_1+R_2$ 2). Each ...
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24 views

Algorithm for uniform sampling with bounded replacement

Is there a simple algorithm to sample from the uniform distribution on sequences of $n$ numbers, each taking one of $m$ integer values from $0$ to $m-1$, where each value can be repeated at most $r$ ...
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54 views

Probability of uniformly drawing N numbers less than the expected second highest value

In the case of 3 draws (N=3) from Uniform[0,1], the expected second highest value would be 1/2. Although unlikely it could happen that all three numbers were less than 1/2. It is exactly this ...
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77 views

Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$

Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y > X, Y \in [-1,1] \}\,?$$ I managed to compute the ...
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Finding the distribution of $\frac{min(X,Y)}{max(X,Y)}$

Just need some hints on finding the distribution of $Z =\frac{min(X,Y)}{max(X,Y)}$ Where X and Y are iid ~ Unif(0,1). $P(Z \gt z) = P(\frac{min(X,Y)}{max(X,Y)} \gt z) = P(min(X,Y) \gt z*max(X,Y))$ ...
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Uniform random variable as sum of two random variables

Taken from Grimmet and Stirzaker: Show that it cannot be the case that $U=X+Y$ where $U$ is uniformly distributed on [0,1] and $X$ and $Y$ are independent and identically distributed. You should not ...
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60 views

estimating the upper bound on a uniform distribution from max order statistic

I have a question. Suppose that $X_1,\ldots,X_n$ are iid $U(0,\lambda)$ and let $X(n)$ denote the nth order statistic. Suppose $\lambda$ is unknown and should be estimated from the sample. Take ...
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97 views

Convergence of $X_{{\lfloor n/3 \rfloor}}^ \space\small{(n)}$ if $X_1, \dotsc , X_n \sim U(0,1)$

$X_1,X_2,\dotsc ,X_n$ are independent, uniformly distributed random variables on the interval $[0,1]$ The question is the convergence of the sequence: $X_{{\lfloor n/3 \rfloor}}^ \space\small{(n)}$. ...
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A question regarding symmetry properties of a uniform distribution [duplicate]

Was anyone able to explain why $$E(U_2) = 0$$ I don't quite understand what the relevance of the underlined statement - "by the symmetry of $U_1$" in determining $E(U_2)$ is edit: I get it now, ...
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UMP test for two different distributions

My question: UMP test for H0: X~u(0,1) vs H1: X~Exp(1) My attempt : By nayman pearson lemma The best critical region is Y >= c Where Y has Irwin-Hall distribution (sum of uniform distribution) and c ...
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Why there is no uniform prior for Box-Cox Power Transformed Normal Models

I am trying to get intuition why uniform prior like below will not work for the box-cox model. Box-cox model: $y^{(\phi)}_i \sim N(\mu, \sigma^{2})$ where $y^{(\phi)}_i = (y^{\phi}_i-1)/\phi $ if ...
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54 views

Expectations of the geometric mean of a random sample from a uniform distribution

If I have a random sample of size n from a Uniform(0,1) and I define the geometric mean as G can anyone give me insight in to how I can find the expected value of G, E[G]? Once I can get my head ...
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26 views

The GPD part of transforming standardized residuals to uniform variates

I need help (or clarity) with transforming a series of standardized residuals, through their parametric CDF, into uniform variates. I started with the awesome R-package, "rugarch": spec <- ...
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Why doesn't runif generate the same result every time?

Why is it that random number generators like runif() in R don't generate the same result every time? For example: ...
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How to I find the distributions for these two random variables?

$X = \min\{U_1, U_2\}$ where $U_1$, $U_2$ are iid ${\rm Unif}(0, 1)$. and $Y = \max\{U_1, U_2\}$ where $U_1$, $U_2$ are iid ${\rm Unif}(0, 1)$.
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Testing uniformity of data

I need to test if a vector of observed values are uniform distribution. Lets assume: This values are not a sample, but my entire universe. I have a dataset of 12000 observations, where most of the ...
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Simulating draws from a Uniform Distribution using draws from a Normal Distribution

I recently purchased a data science interview resource in which one of the probability questions was as follows: Given draws from a normal distribution with known parameters, how can you simulate ...
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validity of MCM proposal distribution? uniform prior?

I am writing a program to compute estimated values. Suppose I I have a prior (discrete) distribution T that I can sample from, but don't know the analytic PMF. That is to say I have a program that ...
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Unifirom distribution from secure random number generator?

I'm testing a Range function from big integer software libraries. The function will return an integer in the range [0,k), where ...
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Distribution of correlation coefficients for uniform random variables

Let $n>1$, let $X$ be uniformly distributed on $[-\frac12,\frac12]$, and consider the sequence $X_1,\ldots,X_{n+1}$ of independent copies of $X$. R implements ...
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Are all sequences of of random (uniform) numbers also uniformly distributed?

If I take some sequences of random numbers generated by a random number generator with uniform distribution, will the resulting sequences be uniformly distributed as well? By example, if I have a ...
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Asymptotic distribution of uniform order statistics

It can be shown that for an iid sample from a Uniform(0, 1) distribution, \begin{equation} n(1-U_{(n)}) \rightarrow exp(1) \\ n(U_{(1)}) \rightarrow exp(1) \end{equation} To see this just try finding ...
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How do I compute the density of this data set that is made up of two different 3D-distributions?

A sequel to this question. I have a dataset where: $\frac{4}{5}$ of the points are drawn from: $(x, y) \sim \mathcal{U}_{2}(0,30)$, $(z) \sim \mathcal{U}_{1}(14.5, 15.5)$. $\frac{1}{5}$ of the ...
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Where is the uniform distribution with one parameter ($U(\theta, k \theta)$) useful for modelling?

I recently came across the distribution $U(\theta, k \theta)$ (where k is known) in the context of statistical theory (as a nice toy example for finding MLE and the likes). However, I was wondering ...
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Scale parameter MLE scheme known but how to find according distribution PDF?

For known location, we can find the scale parameter of a normal distribution by calculating the sum of squared differences to the location, then dividing by n-1 and taking the square root. This is the ...
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Estimating Uniform distribution endpoints using data with errors

Suppose I have a random variable $X$ ~ $Unif(0,\theta)$ where I want to estimate $\theta$. I draw a sample $X_1,...,X_n$.One way is to get a point estimate using e.g. maximum likelihood estimation ...
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Discrete uniform random variable(?) taking all rational values in a closed interval

I just had an (intellectual) panic attack. A continuous random variable that follows a uniform in a closed interval $U(a,b)$: a comfortably familiar statistical concept. A continuous uniform ...
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Correct notation wrt. uniform distribution

Assume that I have a discrete set $L$ and a transformation $\phi: L \rightarrow [0,1]$ that normalizes set $L$ such that now values belonging $L$ are uniformly distributed among the unit interval. ...
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Conditional distribution of uniform random variable distributed over (0,1)

Let $U$ be a random variable uniformly distributed over (0,1). Compute the conditional distribution of U given that $U>a$ The solution says: $P(U > s | U > t) = \frac{P(U > s)}{P(U > ...
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Moment of random variable on a integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
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Self study question on Uniform Distribution

I am attempting a self-study question on uniform distribution but was not able to make sense of the answers provided after taking an attempt. Appreciate any help and guidiance please. Question ...
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65 views

Uniform distribution probability?

Let $X$ and $Y$ be independent random variables, each of which is uniformly distributed between $0$ and $1$. Find the probability that $(X−1/2)^2+(Y−1/2)^2≤1/9$. Give at least $8$ correct digits ...
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48 views

Distribution of MLE of $N$ based on a random sample of size n from discrete uniform dist.(1,2,…,$N$)

Let $X_1, X_2, ..., X_n$ be a random sample from discrete uniform distribution on $(1,2,\ldots,N)$, where $N$ is an unknown positive integer. Find MLE of $N$ and also find the distribution of the MLE. ...
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Calculating probability based on mixed variables

Assume there are $K$ people and iid. parameters $a_1,\ldots,a_K$ associated to them with $a_i \sim U(0,1)$. Person $i$ observes his own fixed $a_i$ with some noise: \begin{equation} X^{(1)}_i= a_i+ ...
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Any practical uses of inverse uniform distribution?

To motivate a paper in game-theory I need examples of real-life uses of the inverse uniform distribution (http://en.wikipedia.org/wiki/Inverse_distribution#Inverse_uniform_distribution). Which type of ...
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Test the randomness (uniformly distributed) on a 64 bit float random generator

We have a random number generator which is supposed to generate 64 bit floats, uniformly. We want to test whether it is a good uniformly random. I am not asking the general way to test it, as it was ...
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PIT on a sample with m bins, and KS test used to estimate a good value for m

I know about PIT, but this works only when you know the distribution, or at least have a strong hint. What I am trying to achieve is to transform a given sample into an equivalent sample with ...
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What is the ratio of uniform and normal distribution?

Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it? I found the ratio of two normals with mean zero is ...
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223 views

Geometric mean of uniform variables

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
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What probability distribution is to the discrete uniform distribution as the beta distribution is to uniform distribution over $[0,1]$?

A beta distribution with its parameters $\alpha = \beta = 1$ is the uniform $[0, 1]$ distribution. What distribution is to the discrete uniform distribution (the sample space is left undecided), as ...
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'Uniformization'?

I am looking for a better term for what I call 'uniformification', where I change data to make it more close to uniformly distributed. I am doing a project in which I try to make the output of a ...
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Expected standard deviation for a sample from a uniform distribution?

I've been trying to find information on the sampling distribution of the standard deviation for uniform distributions and have been having a heck of a time figuring out the expected value for the ...
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Help understanding uniform marginal distribution in Farlie-Morgenstern family.

http://imgur.com/FeFf3e9 The imgur link is to a screenshot of the relevant section in my text. I have trouble understanding how if $H(x, \infty)=F(x)$ is the marginal distribution of $x$, how $F(x) = ...
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243 views

Statistical test for uniform distribution

I have a sample of 5 numbers from known interval [0, 10]. Is 5 numbers is enough to make some conclusions about whether these numbers are drawn from uniform distribution or not?
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48 views

Quality of randomness in generated random number

I have generated a list of 3 random number where each summed to 1. I would like to access the quality of randomness. What is the best mechanism to access this randomness? E.g my random numbers are. ...