Questions tagged [uniform-distribution]

For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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181 views

How to find the pdf of the minimum of absolute differences of Uniform distributions.

Let $X_1$,$X_2$ and $X_3$ are independent random variables that are uniformly distributed over $(0;b), b>0$. What is the probability density function of z=min($Y_1$,$Y_2)$, where $Y_1=|X_1-X_2|$ ...
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1answer
34 views

Uniform distribution- independent random variable max and min

Let $X,Y \sim U_{[-1,1]}$ are independent random variables. Let us define $U=\max (X,Y)$ and $V=\min(X,Y)$. Are random variables $U,V$ independent ?
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112 views

joint/marginal pdfs over parallelogram integration limits

I am TERRIBLE at figuring out the limits of integration when finding PDFs. Say $a, b$ are uniformly distributed over the parallelogram with vertices $(0, 0), (1, 0), (2, 1), (1, 1)$. Find the joint ...
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28 views

Random Area and Perimeter

Lt A and L denote the area and perimeter of a rectangle with length $X$ and height $Y$, such that $X$ and $Y$ are independent, and uniformly distributed on $(0,1)$. Find the density function of $A$ ...
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1answer
52 views

Create random variable Expo from Unif(0,1)

Working through some problems from Introduction to Probability (Blitzstein) Let U~Unif(0,1). Using U, construct X~Expo($\lambda$). My work: (edited with updates on CDF and inverse function) PDF ...
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24 views

Optimal number of experiments

There is a random variable and we know that it is either uniformly distributed on $(0, 1)$ or uniformly distributed on $(0, \frac{1}{2})$. Both cases are equally likely to be. We are to guess the ...
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2answers
82 views

Expected number of terms needed to get a sum greater than $T$, for i.i.d. random variables uniformly distributed in $(0,1)$

Suppose we have $T>0$, and $(X_n)_{n \in \mathbb N}$ is a collection of i.i.d. random variables that are uniformly distributed on $[0,1]$. Define the random variable: $$ N := \max \left\{ n \in \...
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3answers
7k views

Binomial distribution with random parameter uniformly distributed

I have a problem with the following exercise from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001 (page 155, ex. 6): Let $X$ have the ...
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1answer
86 views

PDF of an exponential distribution with varying paramter, lambda

Suppose that the lifetime of a device is exponential with rate λ, but suppose also that the value of λ is not fixed but is itself a random variable that is uniform in the range [a, b) with 0 < a. ...
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1answer
74 views

Independence between random vector and event

Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables. Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables. Consider that $X_i$ ...
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2answers
395 views

Find probability of $P(X Y < 1)$

Let $X$ and $Y$ be two independent $\mathrm{Uniform}(0,2)$ random variables. Find $P(XY < 1)$ I started off by finding the pdf $f_X(x)=\frac{1}{2} $ when $0<x<2$ Same for $Y$. I then ...
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1answer
47 views

For $|G|=m$ and random $x_1, \dots, x_m\in G$, (dis)prove that $\prod x_i$ is uniformly distributed over the elements of $G$.

For $|G|=m$ and random $x_1, \dots, x_m\in G$, (dis)prove that $\prod x_i$ is uniformly distributed over the elements of $G$.
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49 views

Is it possible to get a least squares for the Uniform distribution?

Is it possible to use a Least Square Method for the Uniform distribution? That is, for U~[a,b] is it possible to get LS estimators of a and b? If not, what is the reason?
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41 views

Uniform Distribution Problem on Efficiency

A bus travels between two cities $A$ and $B$ which are $100$ miles apart. If the bus has a breakdown, the distance from the breakdown to city A has a Uniform Distribution over $(0,100)$. There is a ...
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1answer
35 views

Particle in Sphere

The position for a particle is random, with uniform distribution on the sphere that has its centre in origin and radius equal to 7. Calculate the expected value of the particle's distance from the ...
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2answers
22 views

Probability for Uniform distributions

Two people decide to meet at a cafeteria sometime between 12:00 and 13:00. If, each and one of them arrive at random chosen times during the hour, and wait 45 minutes on each other (or until the clock ...
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1answer
47 views

Computing the PDF of $X + Y$ if $X, Y \sim \text{U}(0, 1)$ using a convolution

Theorem: If $X$ and $Y$ are independent continuous random variables with density functions $f_{X}$ and $f_{Y}$, the probability distribution $f_{X + Y}$ of $X + Y$ is given by $$f_{X + Y}(a) = ...
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1answer
56 views

Find the probability that the roots of the quadratic $U_1x^2+U_2x+U_3$ are real

The question, from the textbook: Mathematical Statistics and Data Analysis Let $U_1, U_2, U_3$ be independent random variables uniform on $[0,1]$. Find the probability that the roots of the quadratic ...
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122 views

Probability that Carl has to wait at least 10 minutes for one of the others and for both of the others to show up?

This question is for the other subquestions for the same problem here. For those not willing to click the link, I will post the exercise problem here as well. Alice, Bob, and Carl arrange to meet ...
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2answers
53 views

Symmetry of Uniform Distribution PDF

I'm studying probability theory and came across an exercise problem that I couldn't quite understand, even with the solution, and was hoping someone could give me some insight. Alice, Bob, and Carl ...
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2answers
297 views

PDF of Z = XY for Jointly Uniform (X,Y) with Parabolic Region

"Suppose that $(X,Y)$ is uniformly distributed on the subset of $\; \mathbb R^2$ defined by the inequalities $0 < X < 1$ and $0 < Y < X^2$. Determine the probability density function of ...
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1answer
89 views

Non-monotonic transformation of Uniform distribution and derivative

Let $X\sim Uniform[0,1]$ and $z:[0,1]\rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc). If a function $\alpha$ is defined to be $$\...
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2answers
54 views

Expectation of a battery lifetime: Uniform Distributions

Question: A battery has a lifetime of $24$ hours and it is used for maximum three days. On each day, a person uses the battery for $K$ hours, where $K$ is uniform on $[0,24]$ and independent of the ...
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0answers
107 views

The expectation of a geometric random variable where its parameter is uniform

First thanks for any help editing my text. If a random variable $X$ has a geometric distribution with parameter $P$ where $P$ itself is a random variable and uniformly distributed from $0$ to $1-1/n$,...
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2answers
49 views

CDF of the of Joint Uniform Distribution for k random variables.

If i have K independent Random Variable: $X_1,X_2,x_3,\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot, X_k$ What would be the CDF of the sum of their Joint Distribution? $f_{X_1+X_2+X_3+...+X_k} (z)$ z&...
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1answer
28 views

Generalised (general) Uniform Distribution (continuous)

I have seen the Uniform Distribution/a uniform random variable for some interval in $\mathbb{R}$. For example $U(a,b)$ has probability density function $\frac{1}{b-a}$ (noting this is the 'volume' of ...
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30 views

Uniformly Distributed Marginal Density [duplicate]

A point $(X,Y)$ in the Cartesian plane is uniformly distributed within the unit circle if $X$ and $Y$ have joint density: $$f(x,y) = \begin{cases} \frac{1}{π} & \mathrm{x^2+y^2} \le 1 \\ 0 &...
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30 views

Sum of squares of uniformly distributed variables

Let X, from which I draw N samples, have a uniform distribution on $(-1,1)$. What is the distribution of the sum of squares -- $\sum\limits_{i=0}^{N-1} {x_i}^2$ -- and its variance? We can assume N to ...
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32 views

What is the distribution of a finite sum of products of i.i.d uniformly distributed variables and their indicators

Well , the case is : let $X_i$ ~ $U[0,X_{max}]$ , $i=1,...,N$ - i.i.d random variables from a single uniform distribution. Let $I${$X_i>C$} be an indicator function of $X_i$ ($C \le X_{max}$). So , ...
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3answers
132 views

Distribution of max and min

If $X_1,X_2,\ldots,X_n,\ldots$ are iid uniform random variables on $[-1,1]$. What's the distribution of $X_{\max,n} = \max_{1 \leq i \leq n} X_i$ and $X_{\min,n} = \min_{1 \leq i \leq n} X_i$? My ...
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0answers
34 views

Stochastic Independence of $\tan(U_1)$ and $\tan(U_1+U_2)$ for uniform independent $U_1,U_2$

I found the following statement in Stoyanov's book on counterexamples (Section 7.2) quite interesting: Let $U_1$ and $U_2$ be independent and uniformly distributed on $(0,\pi)$. Then $\tan(U_1)$ and $\...
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1answer
299 views

Average distance between $n$ randomly distributed points on a square with their nearest neighbors

Given a square with side length of $x$ and $n$ randomly distributed points on it with uniform distribution, what is the average of the length between the nodes and their nearest neighbors? I'd be ...
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1answer
61 views

Let (X, Y ) be a random point chosen uniformly on the region R = {(x, y) : |x| + |y| ≤ 1} b. Find the marginal densities of X and Y.

I have graphed the set and realised that x is in [-1,1] and so is y. However I cannot understand what am I to integrate?
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32 views

Don't understand the answer to a uniform distribution question with conditional probability for probability and statistics.

The question is this: You arrive at a bus stop at 10 A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. (a) What is the probability that you will have ...
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1answer
29 views

Drawing a set of unique random numbers < M/2, with N attempts.

We have to draw exactly N random numbers from a random uniform distribution, and want it so that, on repeated experiments, on average half of the unique numbers drawn will be smaller than M/2. The ...
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2answers
39 views

Probability of $x^2 \ge 4$ in uniform distribution

I want to find the probability of $x^2+bx+1=0$ that has at least one real root. Also, $b$ is a uniform random variable on the interval $[-3,3]$. I know the condition for this quadratic equation to ...
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2answers
73 views

Probability of n heads in a uniform distribution

Suppose that $P$ is the probability of landing heads up and also the $P$ variable is a uniform distribution in the interval $[0,1]$. The question is how to find the probability of first $n$ heads in ...
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2answers
299 views

Kurtosis of uniform distribution

I am a beginner in statistics, and am self-studying. I want to determine the kurtosis for uniform distribution. Could someone please help me with this problem?
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1answer
69 views

Uniform distribution on a line segment

a. A point is chosen uniformly at random on a line segment of length L, dividing the segment into two parts. Find the probability that the longer of the two parts is at least twice as long as the ...
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1answer
190 views

transforming exponential distribution to uniform

The question is to find a function that transforms a random variable $X$ that has an exponential distribution given by parameter $\lambda = 1$ such that the function applied to $X$ has a uniform ...
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1answer
157 views

Bates Distribution

Good evening everyone. I might need some help on something. Suppose we have $n$ independent variables from $U[0,1+θ]$ and suppose also that $λ=1+θ$. The estimator of $θ$ is $\bar{θ}=2 \bar{X} - 1$. Ι ...
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2answers
46 views

Suppose 𝑋 and 𝑌 are two independent variables that follow a uniform distribution in [0,1]. calculate $𝑃 (𝑋 + 𝑌> 2)$

Suppose $𝑋$ and $𝑌$ are two independent variables that follow a uniform distribution in [0,1], calculate: a. $𝑃 (𝑋 + 𝑌> 2)$ b. $𝑃(𝑋+𝑌>5𝑋√𝑌)$ Well, i don't get what is this question ...
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1answer
327 views

Expected value and variance of uniform distribution

I have to find the expected value and variance of the uniform distribution X that is 0 for $x<1$ and $x>3$. Also what is the probability $P(x>1.5)$?
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194 views

Uniform distribution, chi square test

What test or procedure can I use to determine the best estimate $\alpha\in [0,1]$ whether given $N$ numbers come from the uniform distribution in the interval $[0,\theta]$ for a given $\theta>0$? I'...
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2answers
26 views

probability of ${Unif(0,1)^2}$

I am new to probability theory. In computer programming, I often use the uniform random number in $(0,1)$ $$ U= Unif(0,1) $$ what is the probability density of $U^2$? In general, how to find the ...
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0answers
32 views

Finding method of moments estimators for uniform distribution.

I am trying to do the following question: I was following Michael Hardy's answer, and I got $\hat{b}=2\bar{x}-\hat{a}$ and the equation $\hat{a}^{2}-2\bar{x}\hat{a}-\frac{3\sum (x_{i}-\bar{x})^{2}}{n}...
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1answer
94 views

Conditional probability with uniform distributions: A company will experience a loss X that is uniformly distributed between 0 and 1

I'm trying to solve the problem: "A company will experience a loss X that is uniformly distributed between 0 and 1. The company pays a bonus to its employees that is uniformly distributed on the ...
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0answers
121 views

Strongly consistent estimator for uniform distribution on $[-\theta, \theta]$

Let $X$ be a random variable having uniform distribution on the segment $[-\theta, \theta]$. I construct the following estimator for unknown parameter $\theta$. $$ \hat{\theta}(x_1,\ldots,x_n) = \frac{...
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2answers
31 views

Conditional expectation of a sequence of random variables

$X_1 \sim \mathrm{Unif}(0,1)$ if $X_1=x_1$, $X_2 \sim \mathrm{Unif}(x_1,x_1+1)$ if $X_2=x_2$, $X_3 \sim \mathrm{Unif}(x_2,x_2+1)$ for $n \geq4$, $X_n$ is defined the same way. How do I calculate $E(...
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1answer
51 views

Calculation of random variable probabilities

I have a question about how to solve this exercise: The monthly sales of a certain consumables store are distributed evenly, with average 4000e and standard deviation 1200e, and the estimated expenses ...