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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47 views

Prove that $E(X_1 \dots X_n)^\frac{1}{n} \leq (EX_1 \dots EX_n)^\frac{1}{n}$, with $X_i$ uniform distribution

Let $X_1, \dots , X_n$ i.i.d. uniformly distributed random variables with $f(x) = 1_{(0,1)}(x)$, $x \in \mathbb{R}$. Let $\Pi_n = (X_1 \dots X_n)^\frac{1}{n}$ and $M_n = \max \{ X_1, \dots , X_n ...
2
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1answer
42 views

Finding the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$

I'm trying to find the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$ Here $U$ is the uniform distribution. The method I use i to introduce an auxilary variable $W=X$ and then use ...
3
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1answer
162 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
4
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2answers
71 views

How to show $\max\{Y_{1},Y_{2},\cdots,Y_{n}\}$ converges in probability to $\theta$ as $n \to \infty$.

Let $Y_{1},Y_{2},\ldots,Y_{n} $ be independent random variables , each uniformly distributed over the interval $(0,\theta)$. Show that $\max\{Y_{1},Y_{2},\ldots,Y_{n}\}$ converges in probability to ...
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1answer
33 views

Simple probability problem.

$X$ ~ $U(0,1)$ and $Y$~U$(0,1)$ are two indenpendent variables. Get Pr ( Y > X). NOW what i don't understant in this problem is how you set the limits of integration. I heard that you must set ...
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1answer
184 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
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1answer
80 views

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 ...
2
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1answer
254 views

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, ...
0
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1answer
324 views

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: ...
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1answer
137 views

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 ...
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1answer
39 views

Weak convergence and limiting distribution

I have $X_{i} \sim \operatorname{Unif}\left(0,1\right)$ iid random variables and have to show that $$ \frac{4\sum_{i=1}^n iX_{i} - n^2}{n^{3/2}}$$ converges weakly and compute its limit. How can I do ...
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2answers
76 views

Find the probability of $ x_2/x_3 \leq a $ where $x_2,x_3$ are uniform i.i.d.

Let $x_1,x_2,...,x_n $ be independent and identically distributed, uniformly on $(0,1)$. How to find $P(x_2/x_3 \leq a)$?
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1answer
191 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n ...
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1answer
68 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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2answers
227 views

Uniform distribution over the unit circle

Suppose that $U$ and $V$ are two independent uniform $(-1,1)$ random variables. Any hints on how I can show that their conditional distribution, given $U^2 +V^2<1$ is given by the uniform ...
2
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1answer
54 views

$(\log n)_{n \in \mathbf{N}}$ not uniformly distributed mod 1.

Let $(x_n)_{n \in \mathbf{N}}$ be a sequence of real numbers, we say that $(x_n)$ is uniformly distributed mod 1 (u.d. mod 1) if $$\lim_{N \to \infty} \frac{|\{1\leq n \leq N : (x_n -\lfloor x_n ...
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1answer
66 views

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 ...
2
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1answer
32 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
83 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
42 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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1answer
31 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
28 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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0answers
66 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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1answer
50 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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1answer
139 views

Probability of maximum of 2 uniform random variables

The random variables X and Y are independent, each with the uniform distribution on [−1, 1]. Find: $$P[max (X,Y) >0.5]$$ Apparently there is an easy approach without integration, but I am having ...
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0answers
34 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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2answers
163 views

Expected value of the function of a uniformly distributed random variable

Let X by a uniformly distributed random variable on the interval [0,1]. Find $E[e^Y]$ I am trying to make use of the formula $$E[g(X)] = \int_{-\infty}^{\infty}g(x)xdx$$ so then $$E[e^X] = \int ...
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0answers
446 views

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
65 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
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0answers
90 views

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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1answer
55 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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48 views

Goodness of fit for uniform distribution

I have a set of $N$ votes $O_1, O_2,..O_N$ distributed into $n$ bins. So... $$n \le N$$ $$0 \le O_i \le N$$ $$\sum_{i=1}^{n} O_i = N$$ I want to generate some sort of metric for how uniformly ...
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4answers
276 views

Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$

What is the probability that $a > b + c$? $a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
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0answers
602 views

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 ...
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0answers
76 views

Uniform distribution

You arrive at a bus stop at 10'0 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. what is the probability that you wait longer than 10 minutes? if at ...
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1answer
18 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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1answer
140 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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1answer
31 views

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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1answer
138 views

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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2answers
55 views

$X$ 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: ...
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2answers
325 views

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 ...
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1answer
187 views

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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1answer
1k 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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1answer
168 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
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1answer
266 views

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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1answer
78 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
40 views

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 ...
2
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0answers
100 views

Expected number on the convex hull [closed]

My question is not very difficult, but I have problem with that : Prove that if S is the set of n points sampled from a uniform distribution in a unit square, then the expected number of points on ...
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1answer
59 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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1answer
154 views

Probability (Statistics) [closed]

The internal sales group of a company has full-time employees on the phone calling prospective customers. Based on historical information, each call only has a 5% probability of being successful. ...