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

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Chi2 test on big uniformly random sample [duplicate]

I expect that, in Python with numpy and scipy, scipy.stats.chisquare(numpy.bincount(numpy.random.randint(100, size=1000000))) will return P-value which is very ...
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Jointly Complete Sufficient Statistics: Uniform(a, b)

Let $\mathbf{X}=x_1, x_2, \dots x_n$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that the ...
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52 views

Finding pdf of transformed variable for uniform distribution

This is from MITx's Intro to Probability and Statistics course, the problem is on this page. Suppose $X \sim \textrm{Uniform}(0,1)$ and $Y=X^3$. Find the pdf for $Y$. Since it's a uniform ...
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Detecting relationships between two sets of circular data

I have a set of points $(x_i,y_i)$ where each x & y value is circular & can take on a value from -pi to pi. (The topology of the data is a torus, but I am not sure how relevant that is to the ...
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Inconsistent results with Monte Carlo solutions to similar problems in probability

I am presently going through the book Fifty Challenging Problems in Probability with Solutions and implementing Monte Carlo solutions to most of the problems in R to get familiar with the language, ...
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2answers
78 views

Expected value of MLE of uniform distribution [closed]

Let $X_1,\dots,X_n$ be a simple random sample from $U(0,\theta)$. Let $\hat\theta=X_{(n)}$ be the MLE estimator. How can I find the expected value of $\hat \theta$ and prove that is it consistency? ...
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Difference between two methods of random point generation [migrated]

In order to do a monte carlo simulation to estimate expected distance between two random points in $n$ dimensional space I discovered the following two similar looking methods to generate random ...
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35 views

Problem involving Scheffe's theorem and asymptotic distribution

If $\{ X_n \}$ are independently and identically distributed $U(0,1)$ random variables and $V_n = n(1 - X_{(n)})$ (where $X_{(n)}$ denotes the $n$th or largest order statistic), then how do I derive ...
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Continuous uniform random variables convergence question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$ How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?
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24 views

setting log-uniform priors in Stan

I have been using Stan for a couple months now and I want to adopt a log-uniform prior on some parameter array real theta[N]. I want to do something like a ...
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Variance of Estimator (uniform distribution)

In my script for statistical signals, I have some troubles to get the same result for the variance of an estimator $T$. Here is the example: Given the observations $X_1, \dots , X_N$ of a uniquely ...
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Markov Chain Monte Carlo (MCMC): How many samples are needed to get a uniform sample?

I am interested in a general answer although my question is rooted in a specific document. I am using the R package "hitandrun": https://cran.r-project.org/web/packages/hitandrun/hitandrun.pdf On ...
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29 views

When is it appropriate to use the Central Limit Theorem?

I am currently having a read through the Statistical Drake Equation; a method of taking the Drake Equation, letting each number be a uniform random variable, and then applying the Central Limit ...
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23 views

Uniform Distribution on (a 0) MLE

If we have X~U(a,0) where a<0 what is maximum likelihood estimation of a ? I tried to find on internet but I could not find any resource about (a,0) interval, there is resources only for (0,a). Is ...
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37 views

Is it possible to use runif() to generate a vector of random variables with different probabilities? [duplicate]

Here's the question I have to solve: Let X be a random variable with pmf as given: P(X = 1) = 0,1 P(X = 2) = 0,3 P(X = 3) = 0,6. Simulate this distribution ...
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Uniform Conjugate prior to raleigh

Can someone please explain why the x0 for the posterior is the max of xi in the below solutions? Thanks
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36 views

Histogram bin size to show deviation from uniform distribution

Simple question: Is there a rule of thumb for number of bins in a histogram with a uniform distribution? Details: I have a stochastic computer simulation that produces, as a test, $n$ values that ...
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1answer
16 views

How to test that sample data is random? [duplicate]

Lets say there's a function that produces a random number between 1-10000. We need to verify that the generated numbers are truly random and that the distribution is uniform. How do we test the ...
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18 views

Statistics Question About Simulating A Dice Roll

A computer programmer is writing a program to simulate a board game. Part of the game involves rolling a fair 6-sided dice. The programmer decides to simulate this roll in the following way. First a ...
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64 views

Uniform distribution MLE

Just a quick question: I know a $U(0, A)$ with density of $1/A$ has as MLE of $X_{max}$, but would a $U(1,1+A)$ have the same MLE that of $X_{max}$? I'm assuming so but just for clarity. Thanks in ...
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19 views

Difference of two Uniform Random Variables [duplicate]

Suppose $A \sim Unif(X, X+1)$ and $B \sim Unif(0,1)$. Find $P(A > B)$. ($A,B$ independent) I started out the following way: $P(A > B) = P(0 > B - A) = P(Z < 0) $, letting $Z = B - A$. ...
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Sum of uniformly distributed random variables over different intervals?

Let $\{X_i\}_{i=1}^N$ be $N$ random variables uniformly distributed over the intervals $[a_i, b_i]$ respectively. How does the sum: $$\sum_{i=1}^N X_i$$ distribute? This is a generalization of the ...
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1answer
29 views

Variance of uniform distribution

I can see how one can compute the variance of a uniform distribution on $[a,b]$ using $$ Var[X] = E[X^2] - E[X]^2 = \frac{(b-a)^2}{12}$$ as explained e.g. here: http://www.statlect.com/probability-...
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1answer
34 views

Bayes theorem with unknown model

I have a book on applied statistics that uses a result I don't understand. The example in the book begins with Bayes' theorem applied to hypothesis $H$ and data $D$: $$P(H|D) = \frac{P(D|H) P(H)}{P(D|...
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Is there an explanation for why there are so many natural phenomena that follow normal distribution?

I think this is a fascinating topic and I do not fully understand it. What law of physics makes so that so many natural phenomena have normal distribution? It would seem more intuitive that they would ...
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Verifying that a random generator outputs a uniform distribution

I asked a student of mine this question: If you have a random number generator that outputs a number between $1$ and $k$, how would you write a test that decides whether the generated ...
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Is there a plateau-shaped distribution?

I am looking for a distribution where the probability density decreases quickly after some point away from the mean, or in my own words a "plateau-shaped distribution". Something in between the ...
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1answer
48 views

Discrete Distribution

In the die-coin experiment, a fair, standard die is rolled and then a fair coin is tossed the number of times showing on the die. Let N denote the die score and Y the number of heads. a)I want to ...
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86 views

Multi-variate uniform distribution

Suppose that $(X,Y,Z)$ is uniformly distributed on $\{ (x,y,z) : 0 \leq x \leq y \leq z \leq 1 \}$ a. I want to find out joint density function of $(X,Y,Z)$. b. I want to find out probability ...
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47 views

How to show $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator

If $X_i \sim$ uniform$(0,\theta)$, how can I show that the estimator $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator for $\theta$? For example, we know the first sample moment ...
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How to find a conditional CDF of a trapezoidal distribution?

I'm working on a game theory model of imperfect information, where players observe certain attributes via noisy signals. Specifically, Player 1 has the opportunity to choose any value $\eta$ from the ...
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1answer
37 views

Expected value of Uniform distribution

Suppose $X$ is an uniform random variable: $X \sim U(a,b)$. I know how to compute $E(X)$, but what if I want to compute: $E(X^\gamma)$ where $\gamma > 0$?
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Consider a random sample of size n from a uniform distribution, $X_i - UNIF(0,θ), θ>0$ and $X_{n:n}$ is the largest order statistic

Consider a random sample of size n from a uniform distribution, $X_i - UNIF(0,θ), θ>0$ and $X_{n:n}$ is the largest order statistic. What is the probability that $(X_{n:n}, 2X_{n:n})$ contains $θ$...
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56 views

Interval of a transformation of a uniform variable

Suppose we have a uniform random variable $U$ which is defined on the $[0,1]$ interval. Consider the transformation:$$X=k\times \log(U)$$ How would I go about calculating the interval on which $X$...
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1answer
29 views

Question on confidence intervals for uniform distribution

I have a random variable $X$ which is distributed uniformly over $[0,b]$. I want to know based on only one observation what is the confidence level of the interval $[X, 4X]$? I know this entails ...
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How to find the CDF of a random variable uniformly distributed around another random variable?

I'm working on some game theory models of incomplete information (which I've posted about a few times here). I think this question is pretty straightforward though, so the actual context is ...
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1answer
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How to compute the CDF of this random variable?

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...
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1answer
39 views

What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
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Why does the number of continuous uniform variables on (0,1) needed for their sum to exceed one have mean $e$?

Let us sum a stream of random variables, $X_i \overset{iid}\sim \mathcal{U}(0,1)$; let $Y$ be the number of terms we need for the total to exceed one, i.e. $Y$ is the smallest number such that $$X_1 +...
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1answer
57 views

Probability the next draw from a distribution is greater than some number given a previous draw

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think ...
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13 views

Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
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79 views

Error in proof of $Y=F(X)$ uniform on $[0,1]$

If we let $Y=F_X(X)$ then $Y\sim U(0,1)$, which is proved in introductory texts in statistics (for example Casella and Berger Statisical Inference p. 54). What is then the error in the "proof" below?...
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MLE Uniform X∈(N,N+1,…,N+10) [duplicate]

I am trying to find N by MLE for several discrete uniform distributions involving a parmeter N∈Z. If the interval X is defined on is X∈(N,N+1,...,N+10) then I think N^=min{X1,X2,...Xn}.But this isn'...
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MLE for discrete uniform distribution [closed]

I am trying to find $N$ by MLE for several discrete uniform distributions involving a parmeter $N\in \mathbb{Z}$. If the interval $X$ is defined on is $X\in (N,N+1,...,N+10)$ then I think $\hat{N}=\...
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Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
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Distribution of continuous uniform RV with upper limit being another continuous uniform RV

If $X \sim U(a, b)$ and $Y \sim U(a, X)$, then can I say that $Y \sim U(a, b)?$ I am talking about continuous uniform distributions with limits $[a, b]$. A proof (or disproof!) will be appreciated.
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Prove that sum of uniform distribution (-1,1) is also uniform (-n,n)? [duplicate]

If $d_i \in U(-1,1)$ (uniform distribution between -1 and 1 - not sure what the canonical notation is for this), then it seems intuitive that $\sum_{i=1}^n d_i \in U(-n,n)$ and thus $$P\big(\sum_{i=1}^...
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1answer
64 views

Relationship between probability distribution and correlation [closed]

I'm unsure of the precise relationship between a probability distribution and correlation, in particular autocorrelation. What exactly is an autocorrelated probability distribution? It seems like ...
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1answer
111 views

chi-squared with too many degrees of freedom

I have a third party random number generator with a period approximately greater than $63*(2^{63} - 1)$ which generates numbers in the range $[0,2^{32}-1]$, ie $2^{32}$ different numbers. I've made ...
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25 views

Test for RVs with known probabilities?

I have written code that generates a sequence of distinct integers. The integers are assumed to occur in the sequence with fixed probabilities. For example, if the sequence contains the numbers [-1,0,...