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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11
votes
0answers
325 views

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 ...
8
votes
3answers
231 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
5
votes
2answers
5k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
5
votes
1answer
332 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 ...
4
votes
2answers
79 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
4
votes
2answers
1k 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| < ...
4
votes
3answers
1k views

Binomial distribution with Uniform parameter

I have a problem with following exercise (it comes from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001, page 155, ex. 6): Let $X$ have the ...
4
votes
2answers
122 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
4
votes
2answers
185 views

Draws from the uniform distribution are taken until the sum exceeds 1. What is the expected value of the final draw?

I was thinking about this question after a related problem: what's the expected number of draws for the sum to exceed 1? For that problem, the answer is known and is a surprising result ...
4
votes
1answer
291 views

The “beach problem”: does anyone know it? or know how to solve it?

The following problem was given some years ago in the German computer-science contest for pupils ("Bundeswettbewerb Informatik"). It was originally wrapped in a story which I will briefly translate ...
4
votes
3answers
120 views

Joint density problem. Two uniform distributions

This is the problem: An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. ...
4
votes
1answer
27 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
4
votes
1answer
132 views

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 ...
4
votes
1answer
37 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
3
votes
4answers
292 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 $\mathbb{R}$.
3
votes
2answers
122 views

Why is $P(X<0)$ the same as $P(X\le 0)$ for continuous distributions?

In uniform distribution, a continuous distribution, for example where $A = -1$ and $B = 1$, $P(X < 0)$ is said to be the same as $P(X \le 0)$. Why is this?
3
votes
1answer
1k views

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 ...
3
votes
2answers
84 views

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

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 ...
3
votes
2answers
73 views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
3
votes
1answer
202 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
3
votes
2answers
1k 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 ...
3
votes
3answers
94 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 $P\{B^{2}-4AC\geq 0\}$. ...
3
votes
1answer
451 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 ...
3
votes
1answer
51 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
3
votes
1answer
28 views

CDF on Standard uniform gives the same distribution

Assume that $X$ has a continuous and strictly increasing CDF $F_X$. Define $Y = F_X^{-1}(U)$ where $U$ is standard Uniform. How dow I show that $X$ and $Y$ have the same distribution?
3
votes
1answer
257 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 ...
3
votes
1answer
757 views

Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
3
votes
1answer
104 views

Circular distribution of circles

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of ...
3
votes
1answer
193 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 ...
3
votes
3answers
464 views

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 ...
3
votes
1answer
147 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 ...
3
votes
0answers
105 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
3
votes
2answers
40 views

Independent two random variables with uniform density

Let $X \sim U[-1,1]$ and $Y=X^2$. Show that $X$ and $Y$ aren't independent. Of course we have $F_X(x) = \frac{x+1}{2}$ and $F_Y(y) = \frac{\sqrt{y}+1}{2}$. But how can I find $$F_{XY}(x,y) = \Pr(X ...
3
votes
0answers
46 views

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 ...
2
votes
1answer
984 views

Probability the three points on a circle will be on the same semi-circle

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ ...
2
votes
2answers
463 views

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece
2
votes
2answers
44 views

Why is this true? (sum of 2 uniform distributions)

If $X\sim U[0,1]$ and $Y\sim U[-1,0]$ and they are independent, then the distribution of $X+Y$ is not simply $U\sim [-1, 1]$, but it is the sum of 2 independent $U\sim [-0.5 ,0.5]$ distributions. Why ...
2
votes
2answers
135 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
2
votes
2answers
47 views

Uniform Distribution Problem

Let $X$ be a random variable uniformly distributed in $[0,1]$, and let $Y$ be a RV uniformly distributed in $[X,1]$. I want to calculate the theoretical distribution of $Y$, any hints? I already tried ...
2
votes
2answers
185 views

$X$ and $Y$ are uniformly ditributed on $(0,1)$. distribution of $\max(X,Y)/\min(X,Y)$

Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution ...
2
votes
1answer
91 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
2
votes
2answers
124 views

Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
2
votes
2answers
109 views

Density of sum of two uniform random variables

I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$. There are many solutions to this on this ...
2
votes
1answer
39 views

Finding the distribution of $M$

Let $(X_1,X_2)$ be uniformly distributed in $[0,1]^2$ and define $Y_1=\max(X_1,X_2)$, $Y_2=\min(X_1,X_2)$. What is then the distribution of $M:=Y_1-Y_2$ ? To find the joint distribution we take a ...
2
votes
3answers
50 views

How do I calculate this expected value?

The problem is as follows: Six players draw, one after another and independently, a number uniformly distributed on $[0,1]$. A player is called a recordist if he draws a number that is larger than ...
2
votes
2answers
85 views

Calc expected value of 5 random number with uniform distribution

Assume we have a random numbers $\sim U(0,100)$. Then the expected value of that number will be: $\int_{0}^{100} \frac{x}{100}$ = 50.5 Now assume we have 5 random numbers $\sim U(0,100)$. How can I ...
2
votes
1answer
33 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 ...
2
votes
1answer
166 views

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 ...
2
votes
1answer
30 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
2
votes
1answer
44 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...