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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Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
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910 views

Limit of sum of (continuous) uniform distributions

In my stats courses at university, I've been working on transformations of distributions etcetera. However, one particular case has intrigued me for a while: the sum of continuous uniform ...
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122 views

how to calculate probability?

I'm facing difficulty in understanding how they in the book, jumped for (12.13) to (12.14). what is given is that $b_1$ and $V_2$ are uniformly distributed between $[0,1]$. I could not post a picture ...
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1answer
2k views

Rao-Blackwell Uniform Distribution

I am having a bit of an argument with my study group about a Rao-Blackwell problem that we have for our statistical theory class. The problem goes like this: Let X~U(0,$\theta$), and suppose we have ...
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1answer
12k views

Uniform Distribution: finding the probability between two variables

Q: In a uniform density $\mathcal{U}(a,b)$ with $a=-0.025$ and $b=0.025$, what is the probability that an error will be between 0.010 and 0.015? A: From the density function, I didn't know how $d$ ...
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541 views

Uniform Probability Distribution

I have a machine part that have lifetime uniformly distributed between 0 years and 1 year. Whenever a part fails, it is immediately replaced with a new identical part. I know that lifetimes of ...
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2answers
32k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniform variables with a difference support, how should one proceed ?...
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1k views

How to estimate parameters of a uniform distribution?

I have information of the order in which students were classified in regard to their scores in a SAT test. I know the distribution of scores for each student is uniform with support [a,b]. I also know ...
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1answer
2k views

Uniform distribution, Expected value and standard deviation for proportion of observations in a subintervall

$X\sim U(0,1)$. Divide the interval [0,1] into k equal subintervals. Then $X_1$=the number of observtions on the first interval. Define the new variable $Y_1=X_1/n$, where n is the number of ...
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1answer
2k views

median of a uniform distribution [0,1]

I need to find the distribution of the median from the given distribution, where n is known to be odd. The formula given in class for this is: $n=2m+1$ where $m\in\mathbb{N}$ $f_{x(m+1)}(x)=\frac{(...
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2answers
944 views

Comparison of 2 samples from different uniform distributions

Given that $0\le a\le b<1$ and $p$ is uniform on $[a,1]$ and $q$ is uniform on $[b,1]$ then if $p$ and $q$ are random selections then what is the probability that $q>p$? Edit: I am trying to ...
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344 views

Is transfert theorem the best choice in this kind of exercise?

I am studying Probability theory and came to this exercise : Let $U,V$ be independent uniform random variables over $[0,1]$. Show that $X:=\cos(2\pi V)\sqrt{-2\ln U}$ and $Y:=\sin(2\pi V)\sqrt{-2\ln ...
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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
157 views

Uniform distribution

$X~U(0,1)$ Find $E[\sqrt {X}]$ and the probability density function of $Y$ defined as $Y=X^2$. I know that $E[Y]=E[g(X)] = \int_a^b g(x)f(x) dx$ I don't know why this question does not meet the ...
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112 views

Finding Limits of Integration

I have two functions, one depending on $x$ which is $\frac {1} {2} {\delta(x-5)} + \frac {1} {4}$ which is the combination of a dirac delta function at $5$ and a uniform distribution from $5$ to $7$. ...
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Expected number of overlaps between intervals

Suppose $N$ intervals of length $\delta$ are positioned in $[0,1]$. The starting point $l_i$ of each interval is drawn from an uniform distribution, i.e., $l_i \in [0, 1-\delta]$, thus it will ...
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2k views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
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82 views

Distribution of binary digits in moduli

Considering the (infinite) set of all positive integers that are a product of $2$ primes only, represented in binary $100...01$. Question: is the distribution of the proportion of $0,1$ digits "...
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1answer
74 views

Consider $X \sim \text{Unif} (\alpha, \beta)$. Find $P(X<\alpha + p(\beta - \alpha))$ Assume $p$ is a constant with $0<p<1$

Consider $X \sim \text{Unif} (\alpha, \beta)$. Find $P(X<\alpha + p(\beta - \alpha))$ Assume $p$ is a constant with $0<p<1$
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2k views

Finding the mean of a uniform distribution?

I have a random set $\{a,b,c\}$ and a second set $\{e,d\}$ I draw one number first number and one from the second Letting $X_1$ denote the first number and $X_2$ the second number find, $E(X_1)$ and $...
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6k views

Prove there exists no uniform distribution on a countable and infinite set.

Can anyone help me with this problem, I can't figure out how to solve it... Let $X$ be a random variable which can take an infinite and countable set of values. Prove that $X$ cannot be uniformly ...
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234 views

uniformly distributed question…

I am doing self-study. but I don`t calculate this problem. so I want to calculating course. Three numbers are randomly selected and rounded-off to the nearest interger. Let $X_1, X_2, X_3$ denote the ...
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1answer
319 views

Distribution probability of elements and pair-wise differences in a sorted list

Suppose a set of $m$ integers from $0$ to $n-1$. The integers are uniformly distributed and unique in the set ($n \gg m$). Then, put all the integers into a list an sort that list: $$x_0 < x_1 < ...
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1answer
2k views

Order statistic and independence

"If $Y_1, Y_2, ..., Y_n$ are independent, uniformly distributed random variables on the interval $[0, \theta]$, show that $U=Y_{(1)}/Y_{(n)}$ and $Y_{(n)}$ are independent." I have already found $$f_{...
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2answers
348 views

Prove that a function is uniformly continuous

This is my question : Let $f$ be defined on an interval $I$, and suppose there exists an $M>0$ and $\alpha>0$ such that $$ |f(x) - f(y)| \leq M|x -y|^\alpha, $$ for $x,y \in I$. Prove that $f$ ...
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614 views

Conditional expectation of $X$ given $\sin(X)$

What is the conditional expectation of $\mathbb{E}(X\mid\sin(X))$ if $X$ is uniformly distributed on $[0,\pi]$? Intuitively I expect that it is constant and equal to $\frac{\pi}{2}$, since the Borel ...
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2answers
2k views

Sum of random variables uniformly distributed (0,1) and (0,2)

I'm trying to get $P(0.9<Y<=1.8)$ for the sum of 2 random and uniform values x1,x2 (so that y=x1+x2) where $x1$~$u(0,1)$ and $x2$~$(0,2)$ and I'm trying to do the convolution for it. Seems like ...
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distribution questions

Let $X$ have a uniform distribution $\operatorname{U}(0,1)$, and let $Y = a + (b-a)X,\, a < b$. (a) Find the distribution function of $Y$. (b) How is $Y$ distributed? Let $X_1$ , $X_2$ be ...
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1answer
60 views

Fetch Minimal Record with Probability

I have three records. the records means intervals, $A: [1, 5]$ $B: [2, 6]$ $C: [4, 6]$ A, B and C are three humidity sensors. Value of A is between 1 to 5, B is between 2 to 6, and C is between 4 ...
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1answer
191 views

Measure the uniformity of distribution of points in a 2D square

I am currently running into this problem: I have a 2D square, and have a set of points inside it, say, 1000 points. I need a way to see if the distribution of points inside the square are spread out (...
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Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...
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1answer
579 views

Uniform distribution of points on the surface of a circle around a randomly chosen point

In a Monte Carlo simulation i have encountered the following problem: given a unity vector u defining a point A on the surface of a unity sphere, i must randomly determine a new vector forming an ...
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2answers
76 views

Independence of 2 variables derived from dice rolls

A while ago, I was told that if you roll a d%, by rolling 2 ten sided dice and treating one as the tens digit and the other as the ones, then you can gain 2 independently distributed values by ...
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1answer
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Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
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1answer
295 views

Probability that two numbers do not follow each other and are distributed over a sequence

Assume a sequence $S$ of numbers out of the set $N={1..n}$. Example: $$S = "123312"$$ Set of all pairs would be: $$M = (2,3),(3,3), (3,1)$$ Not in $M$: $(1,1)$ : not occuring in the sequence next ...
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Calculating probabilities in two overlapping continuous uniform random variables (with an added constraint)

Given X~unif(a, b) and Y~unif(c, d) with a < c < b < d. What's the probability that Y>X and Y being realized in the interval (c, b)?
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181 views

Whats the probability that a set of dunif random variables is strictly ordered?

Let $\{X_i\}$ be $n$ iid dunif(0, u) (discrete uniform) random variables with u>n. How do I compute the probability that $\{X_{i+1}\}$ > $\{X_i\}$ for all i?
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1answer
235 views

Very Important question. Limiting distribution

I have an exam in the morning and there is still one question I cannot do. $X_1, \ldots, X_n$ are iid random variables each having distribution with density $f_{X_i}(x;\theta)= 1/\theta$, for $x \...
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1answer
74 views

Generating samples from $u(7,10)$

I have the following assignment: It requires to generate samples from $u(7,10)$,the uniform distribution on the interval $2 \leq x \leq 11$. Compare the normalized histogram with the density function(...
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406 views

Average sine of an angle between two rays in a cone

I'm looking for an average value of sine of an angle between two rays, lying within a cone with a certain angle. Given a cone with an aperture of ${2\chi}$ and two rays lying within the cone. The ...
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1answer
49 views

Uniform Continuous R.V. - Optimization

working on this problem: A road construction company needs to decide where to place an emergency phone on a stretch of road of length L. Suppose that accidents can happen uniformly at random on ...
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1answer
7k views

Continuous Random Variable - Uniform Median, Exponential Mode

Working on this question: The median of a continuous random variable with CDF $F(x)$ is the value $m$ that guarantees that $$P\{X > m\} = P\{X < m\} = \frac{1}{2}$$ The mode is the ...
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1answer
610 views

Normal Random Variable - uniform distribution

So here's the question I'm trying to solve: A stock price movement model supposes that if the current stock price is s, then, after one period, the stock price will be $us$ with probability $p$ ...
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1answer
2k views

Uniform Random Quaternion In a restricted angle range

I'm trying to sample uniform random rotations. I'd like the rotations to be restricted in a range [-θ,θ]. I found a method by K. Shoemake which can be summarized as: ...
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117 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...
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1answer
169 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...
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1answer
1k views

PDF/CDF and expected value of a function

How can I compute the PDF/CDF and expected value of the following function: $$ \frac{\alpha}{r^2} $$ where $r$ is generated as follows: draw $x$ and $y$ from a uniform distribution in the range $[...
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1answer
606 views

Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value

Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $c$? I've messed ...
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2answers
4k views

Probability of difference of random variables

How can I compute this probability? I do not know what to do since it involves two random variables. Let $X$ and $Y$ be uniform random variables on $(0,1)$. How can I compute this? $$ P(|X-Y| < ...
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1answer
1k views

Rank of a random variable that follows a Uniform Distribution (0,1)

Well the question is a little easier .. Let X be a random variable that follows a Uniform distribution (0,1)(Uniform Standard). What is rank of the variable? (Values ​​can take). I have a confusion ...