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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Expectation and probability [closed]

A point P is taken at random in a line AB of length 2a, all positions of the point being equally likely. Show the expected value of the area of the rectangular AP.AB is $2a^{2}/3$ and probability of ...
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minimum distance estimation

I need to set up a minimum distance estimator for the uniform distribution $U[0,\theta]$ and take the $\mathcal{X}^{2}$ statistic as distance https://en.wikipedia.org/wiki/Minimum-distance_estimation $...
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Marginal distribution of uniform distribution over sphere

Let $(x_1,…,x_n)$ be a random vector uniformly distributed on the $n$-dimensional unit sphere. Is there a closed form solution for the joint distribution of $P(x_1, x_2)$?
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Cumulative Distribution Function of $S_{N_{t}}$ where $S_{N_{t}}$ is the time of the last arrival in $[0, t]$

I am confused on this problem. My professor gave this as the solution: $S_{N_{T}}$ is the time of the last arrival in $[0, t]$. For $0 < x \leq t, P(S_{N_{T}} \leq x) \sum_{k=0}^{\infty} P(S_{N_{T}}...
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Posterior distribution of two i.i.d. uniform r.v. given their difference with graphical intuition

I have two i.i.d. random variables, $\theta_1$ and $\theta_2$ which are uniformly distributed on the unit square. I need to compute the joint posterior distribution of these two variables, given their ...
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How to standardize uniform variates?

How do you standardize a set of uniform variates on the interval (0,1) to have mean 1/2 and variance 1/12, while staying in the interval (0,1)? The usual procedure of shifting and scaling variates to ...
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Solving for expected value of the sum of 2 random variables

Let $X_1$ be the time between a road accident and the action of calling the ambulance. Let $X_2$ be the time between calling the ambulance and the ambulance arriving. Suppose the joint pdf of $X_1$ ...
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Show that the maximum of $x_1,…,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. (From definition)

Problem Show that the maximum of $x_1,...,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. Background This question has been asked before, but most answers tackle the ...
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1answer
51 views

Number of samples required to have visited all classes of a uniformly distributed distribution?

So let's say I have a set of binary vectors $x \in \{0,1\}^n$. Hence, $|\{x\}| = 2^n$ . For all $x$, there is a class $c_i$. We do not know what is this class a priori, but we can compute it once we ...
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Suppose X1∼U(0,1) and X 2|X1 =x1 ∼U(0,x1) are uniform random variables. Compute probability of (X1+X2≥1)

The answer to this problem is (1-ln2). I am getting 0.5 which is not even close. Any kind of hints or even suggestive reading would be helpful as I am getting a lot of doubt in problems of the same ...
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Bayes continuous case and a uniform marginal density

Assume I have a population $A$ distrubuted Uniform $U(a, b)$ and now assume I have a Conditional probability distribution conditioned on $S$ given $A$ i.e. $f(S | A)$ where $S$ is a binary variable ...
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27 views

Uniform distribution and probability

Let $Y \sim \mathcal{U}(0,4)$. If 20 independent random samples are extracted, what is the probability that in at least 5 of them $Y > 2$? My attempt was: the required probability should be given ...
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48 views

Finding the posterior with a uniform prior

Suppose I have a normal distribution $X_i \sim N(\mu, \sigma)$ and I impose a uniform prior on the mean, so that it has to be positive. How do I go about finding the posterior distribution $p(\mu | ...
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3 uniform points on a circle

Suppose 3 (distinct) points are uniformly and independently distributed on a circle of unit length (smaller than a unit circle!). This is really circle and not disc. Call one of these points $B$. Let $...
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CDF of $M(X,Y)$?

Let $X,Y \sim^{\text{iid}} Unif(0,1)$. Let $M = M(X,Y) = \min\{X,Y,1-Y,1-X,|X-Y|,1-|X-Y|\}$. Supposedly $image(M) \subseteq (0,\frac13)$ and distribution of $M$ is $F_M(m)= (3m(2-3m))1_{(0,\frac13)}(m)...
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Confidence interval for the mean of the uniform distribution

I can take samples of a random variable $X \sim U(a, b)$, where the length of $(a, b)$ is known. I am interested in its mean $E[X]$, estimated with $\hat{X_n} = \frac{1}{n} \sum_{i=1}X_n$, but I need ...
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Independent random sampling without replacement for sampled Gaussian mechanism

I was reading a paper about "Sampled Gaussian Mechanism" which is used in training differentially private ML models. This mechanism is a composition of ...
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Getting two different results in Covariance Calculation

Let $A=\{(x, y) \in \mathbb{R}^{2}: x^{2}-\frac{1}{2 \sqrt{\pi}}<y<x^{2}+\frac{1}{2 \sqrt{\pi}}\}$ and let the joint probability density function of $(X,Y)$ $f(x, y)=\begin{cases} e^{-(x-1)^{2}},...
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Mean number of throws to exceed a threshold [duplicate]

Say that you have a die with n faces, and you need to throw the die until the sum of your results exceeds a given threshold. What is the average number of throws needed? I think that to compute that ...
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1answer
36 views

Proving the convergence of the maximum of Uniform Distribution

I have a random sample of size $X_1, X_2, .., X_n$ following $U(0,2)$. I need to prove that $X_{(n)}$ which is the maximum ordered statistics will converge to $2$ in probability and almost surely. I ...
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Distribution of ratio of minimum and maximum of uniform

I have a problem in which $X_1, X_2, .., X_n$ follows $U(0, \theta)$. We have $X_{(n)}$ as the maximum and $X_{(1)}$ as the minimum. I am required to compute the correlation between $X_{(n)}$ and the ...
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How to choose priors for experimental data

My question results from the subjectivity of priors, and if there are bodies of work that help to create a more objective approach towards prior choices. My question specifically is to do in the realm ...
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1answer
35 views

Where is the error?

I am trying to compute expectation of $X\mathbb I_{[X+Y\le a]}$ where $a$ is a fixed positive integer, $X$ is discrete uniform random variable taking values from $1$ to $a$, and $Y$ another random ...
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Integrating an absolute difference transformation

I have a dependent variable which is $$ W = |X - Y| $$ X and Y are independently distributed where $ X \sim Uniform(0,1)$ and $ Y \sim Uniform(0,2)$ What am I supposed to do is find the probability of ...
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1answer
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Confusion in deriving the distribution of sum of two uniform variates

Given X and Y as two independent random variables following $U(0,1)$ and We are required to obtain the distribution of $Z = X + Y$. The answer is given as follows: $$f(z) = \begin{cases} z & \text{...
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Probability of difference for uniform sample of consecutive numbers [duplicate]

Lets assume I have a set N which consists of all natural numbers from 1 to n. Now I take a uniform sample of size S from this set. Intuitively I wonder how large the probability is to have a gap ...
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Finding the copula that obeys a sum constraint [duplicate]

I am wanting to sample from a distribution with uniform marginal distributions but with an interesting joint structure. Let the random vector be denoted $X = (X_1, X_2, \ldots, X_n)$ with $X_i \sim \...
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How do I calculate the probability density function for a joint beta + uniform distribution? [duplicate]

The PDF of a Beta distribution is $$ f_X(x) = {{x^{a-1}(1-x)^{b-1}}\over {B(a,b)}} $$ and CDF $$ F_X(x) = I_x(a,b) $$ The PDF of a uniform distribution is $$ f_Y(y) = {1 \over {m-n}} \space for \...
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How do I calculate the probability density function for a joint beta + uniform distribution?

The PDF of a Beta distribution is $$ f_X(x) = {{x^{a-1}(1-x)^{b-1}}\over {B(a,b)}} $$ and CDF $$ F_X(x) = I_x(a,b) $$ The PDF of a uniform distribution is $$ f_Y(y) = {1 \over {b-a}} \space for \...
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Expectation of uniform variates

Let $X_{1},X_{2},X_{3}$ be random variates from $U(0,1)$. It is required to compute $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}})$. Here is what I did.. $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}}) = E(1 ...
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Expectation of conditional uniform variates [duplicate]

I have two random variables $X_{1}, X{2}$ following $U(0,1)$. I need to compute $E(X_{1}|X_{1} > X_{2})$. I am thinking that since the random variables are independent, so $X_{1}|X_{1} > X_{2}$ ...
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Ratio of difference and sum of i.i.d. exponential random variables

I just noticed that for two i.i.d. exponential random variables $X$ and $Y$, the combination $\frac{X-Y}{X+Y}$ appears to be distributed uniformly on $[-1, 1]$ (ignoring the case $X=0, Y=0$ for the ...
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Noninformative prior distribution: uniform or normal? [closed]

The uniform distribution, with the support that has a finite measure, guarantees that the entropy is maximum(as stated in this answer), but in our daily life, normal distribution seems more ...
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Trouble finding var(ax)

So the variance of a 6-sided (1,2,3,4,5,6) die is $291.6$ using the formula: $$ \text{Var}(X) = \frac{(b-a+1)^2}{12} $$ Also, $\text{Var}(10X) = 10^2 \cdot \text{Var}(X)$, so that would mean $\text{...
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If X and Y ~ Uniform(0,1), what is the distribution X/Y? [duplicate]

Given that $X$ and $Y$ are random variables drawn from a uniform distribution between $0$ to $1$, i.e. $$ X \sim Uniform(0, 1) \\ Y \sim Uniform(0, 1)$$ And given that $Y \neq 0$, what do we know ...
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Mean of uniform two-dimensional probability density function

I am trying to calculate the mean of a two-dimensional probability density function, which looks like: and is defined by I know that I can calculate this by However, this is where I get stuck, as I ...
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1answer
113 views

minimal sufficient statistic for $U(\theta, \theta+c)$. $(\theta,c)$ unknown

Suppose $X_1,\cdots,X_n$ are $i.i.d$ from a distribution with p.d.f $$\delta_{(\theta,c)}(x)=\frac{1}{c}\mathbb{1}_{(x\in[\theta,\theta+c])},$$ where $\theta\in\mathbb{R}$ and $c\in\mathbb{R}^+$ ...
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How to find the average distance between randomly distributed points in a rectangle?

Assume there are n points randomly distributed in a rectangle (x being the height y being the width) shown below in the figure. I would like to calculate the average distance between 2 random red ...
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2-dimensional minimal sufficient statistic for $U(-k\theta+k,k\theta+k)$

Find a two dimensional minimal sufficient statistic for $\theta$ from $n$ independent random variables $X_k\sim > U(-k\theta+k,k\theta+k)$, $k\in\{1,\cdots,n\}$ Here is what I've attempted. The ...
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Proving two statistics are equivalent

I was trying this problem from the book - Theory of Point Estimation by Lehmann and Casella. I could do part $a$ and part $b$ but couldn't do part $c$. Any help is appreciated. Just to clarify $T(x)$ ...
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Testing uniform distribution of a discrete function derived from a random number generator

(Forewarning: I'm a programmer, not a statistician, so I apologize in advance for any misuse of terminology!) I'm testing a known random number generator that implements the PCG algorithm. This RNG ...
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Can one uniformly generate complex numbers of absolute value less than a given constant $R \neq 1$? [duplicate]

Can one uniformly generate complex numbers of absolute value less than a given constant R? This would appear to be equivalent to picking points $(x,y)$ uniformly in a disk of radius R, where $x$ is ...
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1answer
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Optimal choice from uniform distribution

I found the following question for a quantitative research role on the website of a trading firm. I tried solving it and I think the answer is 0.5 for all three questions. I think so because both the ...
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Generate uniform distribution under multiple constraints

I have to acknowledge that my skill in statistics are really rusted. I would like to implement in Python a uniform distribution that satisfies constraints on the mean, the median, the standard ...
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1answer
25 views

likelihood function with uniform prior

Assume I have a data sequence {y1,y2,y3,...}. The data comes from y_t = 1 - gamma * t + epsilon_t where ...
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31 views

Conditional pdf of a product of two uniform independent variables [closed]

Let A ~ U(0,1) and B ~ U(0,1) be independent uniform random variables and T = AB, the product of A and B. Find the conditional pdf f(t| B = 2/3) and the joint pdf of A and T f(a,t). Started by finding ...
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55 views

Bayes Estimate for Mean Squared Loss in Uniform Prior

Can some one please help me out in Verifying if my prior distribution is uniform then will my Bayes estimate will always be MLE or UMVUE? If $X_i$ follow iid $N(\theta,1)$ and prior distribution of $\...
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1answer
25 views

Remove points, but preserve diversity

Given a finite set of points in $\mathbb{R}^n$, all at finite distances from each other I would like to remove a certain proportion of points, in such a way as to penalise clustering. For example for $...
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54 views

Order statistics of independent but non-identical uniform distribution

Suppose we have independently distributed $X_i \sim \text{Uniform}(0,a_i)$ where the $a_i>0$ are fixed numbers. I want to obtain the probability that $X_j=X_{n-i+1,n}$ where $X_{n-i+1,n}$ is the $n-...
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52 views

How many unique values can you expect after throwing a dice with k sides?

I ended up asking here. My Problem might be familiar with the Coupon collector's problem and related to this post Probability of throwing n different numbers in m throws of a dice but it does not ...

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