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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Finding density function for uniform distribution

Can anyone help me set this up correctly, please: John is going to eat at at McDonald's. The time of his arrival is uniformly distributed between 6PM and 7PM and it takes him 15 minutes to eat. Mary ...
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Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: http://en.wikipedia.org/wiki/Irwin%E2%80%...
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Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, $...
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Density of uniformly chosen random point inside triangle

Imagine the triangle inside of the points $(0,0), (0,1)$ and $(1,0).$ Let $(X,Y)$ be a uniformly chosen random point from the triangle. Then find the joint density of $X$ and $Y$. The answer is $f_{...
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459 views

Bivariate and Multivariate Probability Distributions

For my homework for Bivariate and Multivariate Probability Distributions section, I encounter the terms joint density, joint distributed random variable, joint probability, uniform distribution, when ...
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Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only one ...
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1answer
52 views

Looking for a simple bivarate uniform distribution with non-zero covariance matrix

Obviously there are many forms this can take, I'm looking for on that gives an non-zero (off diagonal elements) covariance matrix. Does anyone know of one?
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87 views

Probability that at least one event (out of two uniform RV) happens before two other random events

I recently faced a probability problem that is puzzling me. I would like to ask you if you could help me. I have two random variables X1 and X2 i.i.d with uniform distribution U[64,96] and other two ...
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1answer
45 views

Uniformly Distributed ingredients

Suppose we need to make a dish that has three ingredients A, B and C. All are distributed uniformly between [0, 2], [0, 2], [0, 1] respectively. To create the dish, we need 1/4 of A, 1/4 of B and 1/8 ...
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1answer
234 views

Finding the joint density of two random variables

Suppose (X,Y) is uniformly distributed over the region { (x, y) : 0 < x < y < 1 }. Find the joint density of (X, Y). I started out by drawing the unit square and filling in the area where 0 &...
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177 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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396 views

Finding probability of uniform random variable given a condition with another random variable

Suppose X and Y are independent and uniformly distributed on the unit interval (0,1). Find: $$P[Y>\frac{1}{2}\,|\,Y>1-2X]$$ How I approached it was to find the area where $Y > 1 - 2X$, and ...
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48 views

choose a list of words such that have equal letter frequency

I have a big list meaning full Words. surely letter frequency of this word list is different for each letter. Now my problem is to find a way to randomly select words from this word list to a new ...
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What is the expected time you have to wait until the first bus comes?

 three buses, bus A, B, and C come to a bus stop every hour. The time at which each bus arrives at the stop is distributed as a uniform random variable, i.e., TA,TB,TC ∼ Unif[0,1] hours. The ...
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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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Summing many non-standard i.i.d. uniform random variables

all! I have looked up a fair bit on this question and learned much about the problem. But haven't been able to get any crisp answers. Sorry, if I'm missing something obvious. I know one can use the ...
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80 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
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Finding a PMF with random variables

Let $X$ be a discrete random variable that is uniformly distributed over the set of integers in the range $[a,b],$ where $a$ and $b$ are integers with $a<0<b$. Find the PMF of the random ...
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572 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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Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition above ...
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1answer
36 views

What is the distribution of the impact point of a Random Ray

In the $\displaystyle (O,x,y)$ plane, a random ray emerges from a light source at the point $\displaystyle (-1,0)$, towards the $\displaystyle (O,y)$ axis. The angle with the $\displaystyle (O,x)$ ...
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802 views

Linear Combination of Min and Max of Uniform Random Variables

Let $p$ and $q$ be uniformly distributed on $[0,1]$. Define $x=\min\{p,q\}$, $y=1-\max\{p,q\}$ and $z=1-x-y$. What are the distribution functions of $x$,$y$ and $z$? I've got $F_X(x) = 1 - (1-x)^2$ ...
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Finding variance .

Suppose that $f : [0, 1] → [0, 1]$ and we wish to estimate $$I = \int_{0}^{1} f(x) dx$$ Using the hit-and-miss method, we obtain the estimate $$\hat I_{HM}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ where $...
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Plot the cdf and simulate a random variable (rv) with this cdf using the inversion method.

Consider the continuous random variable with pdf given by: $$f(x) = 2(x − 1)^2;\quad 1 < x ≤ 2$$ $$f(x) = 0;\quad \text{otherwise}$$ Plot the cdf for this random variable. Show how to simulate ...
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$X$ is half normal and $S ∼ U{(−1, +1)}$. How $Z = SX ∼ N(0, 1)$?

If we chop a standard normal distribution in half and use only the positive side (scaled up by a factor of $2$ to maintain a proper density), then we get the so-called ‘half normal’ density: $$f_X(x)=...
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Uniform Distribution : pdf & inverse cdf

$X\sim U(1,3)$. Verify that X has cdf $F_X(x) = 2(x − 1)$ for $x \epsilon(1, 3)$ and thus that $F^{−1}_X (y) = 2y +1$ for $y \epsilon (0, 1)$. My attempt for $F_X(x)=\int_{-\infty}^{\infty}\frac{1}{b-...
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Question regarding finding Joint distribution of two random variables

I have a question regarding finding the following joint distribution. Let $p \sim U[0,1]$, standard uniform distribution. The random variable $X$ is defined as $X = 2$ with probability $p$ and $X =...
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8k views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...
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188 views

spin arrow of random variables

Spin an arrow attached to the center of a circular board, let theta be the final angle of the arrow, theta<= 2pi. The probability that theta falls in a subinterval (0, 2pi] is proportional to ...
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1answer
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On uniform number generation with vectors

Let $\vec{a}$ be a random unitary vector. If $\vec{\lambda}$ is a uniformly distributed vector on $\mathbb{S}_2$ (the unitary sphere?), could we say that the result $|\vec{a}.\vec{\lambda}|$ is ...
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1answer
107 views

Equivalence of uniform distribution

Behind a rectangle grid evenly (i.e. uniform distribution) scattered dots. Could it be considered identical (will have the same uniform distribution) to a sequence of independent events with ...
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2answers
54 views

Number of times you have to make a bet on a uniform distribution to expect to achieve a minimal result

Edited for the sake of clarity: If you have a random variable $Q$ distributed uniformly on some interval, say $[a,b]$, what is the function $f$ that describes how many times you have to draw on the ...
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How Do I Get This Joint Density Function?

Given $X \sim u(0,1)$, we define $Y=1-X$, then we have that $f_{X}(x)=I_{[0,1]}(x)$ and $F_{X}(x)=xI_{[0,1]}(x) + I_{(1, \infty)}(x)$. I know, if $0\le y \le 1$ $$F_{Y}(y)=P[Y \le y]=P[1-X \le y]=P[1-...
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Is $e$ uniformly distributed in all bases?

There has been talk of whether or not $\pi$ is normal, i.e. uniformly distributed in all bases $b$ where $b\ge2$. The general response has been that we expect that it is, and have found no obvious ...
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Max variance of uniform distribution?

Suppose I roll a 20-sided die 1000 times and count the number of times a particular value comes up. This gives an array of 20 counts, and the expected value of each is 1000/20 = 50. I'd like to find ...
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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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Random Variables from $[0,1]$ - Integration Limits

I was wondering if someone could help me understand the first steps I should take for solving the next problem: Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
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335 views

Sum of Two (General) Uniform Random Variables

I would like to know if there is a general formula for calculating the sum of two uniform distributions $X$ and $Y$, where $X$ is uniformly distributed on $(-a,a)$ and $Y$ is uniformly distributed on $...
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Is $X_i$ in the following question uniform $\,k$-wise independent bits?

This is a homework question in the book named probability and computing. $13.9$ : suppose we are given m vectors $\overrightarrow v_1, \overrightarrow v_2 , \...
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830 views

Generating points in rectangle

I am trying to generate $N$ points randomly and uniformly distributed in an $m \times n$ rectangle. How can this be done? I have tried to split the initial rectangle into as many rectangles i could, ...
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calculate probabilty, Uniform distribution

this is my first question so excuse my unknowing and mistakes: I was reading a book and just faced this thing: (1.4) $=P(X\gt Z/2)(Y-X)$ (1.5) $=P(2X\gt Z)(Y-X)$ (1.6) $=\min\{{2X,1\}}(Y-X)$ ...
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A Problem on Uniform Probability Distribution

Consider three independent uniformly distributed (taking values between 0 and 1) random variables. What is the probability that the middle of the three values (between the lowest and the highest value)...
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Why do prime numbers in modulo result in more uniform distributions?

Let us assume a sequence as follows: $S_{n} = (S_{n-1} * c_{1} + c_{2})\text{ mod } m$ This is the pseudorandom generator found in most programming languages' random function. It is known that a ...
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Showing the self energies of $N$ uniformly charged disks is proportional to $N^{3/2}$

How would I go about doing this? I assume it is some integral I have to solve, but I have no idea what. (Note:Not a physicist so please excuse incompetence with regard standard notation.) Context ...
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What is the difference between these two questions?

Consider two independent random variables: X~U(0,1) and Y~U(0,2). Let Z = min(X,Y). b) Find F$_Z$(z) in terms of F$_X$(.) and F$_Y$(.). c) Eliminate F$_X$(.) and F$_Y$(.) to find F$_Z$(z). What is ...
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202 views

Probability of elements in a subset of the original set

Let me try and rephrase the question as an example. I'll use bits since its convenient in this case. You have 3 bits A, B and C, that have probability 1/2 of being ...
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In this Cumulative Distribution Function, am I finding the wrong term?

Question I was given: Let V be a uniform random variable distributed over the interval (0,1). Let $\ X = \frac{1}{\sqrt(U)}$. What is the cumulative distribution function and probability density ...
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892 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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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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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 ...