2
votes
1answer
79 views

Uniform distribution with random support

I have a uniform distribution $ X \sim U(A,B) $ where the limits themselves are random: $A \sim N(\mu_A,\sigma_A^2)$ and $B \sim N(\mu_B,\sigma_B^2)$. Hence the support of $X$ is random. $A$, $B$ are ...
1
vote
0answers
45 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
4
votes
2answers
95 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 ...
0
votes
1answer
34 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
3
votes
1answer
176 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 ...
0
votes
0answers
18 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...
2
votes
1answer
38 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 ...
1
vote
0answers
23 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
4
votes
3answers
54 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]. ...
1
vote
1answer
77 views

Convolution of two Uniformly distributed r.v. ove

Assume a continuous random variable $X$ that is uniformly distributed $\underline{\text{on}}$ a $k$-sphere. For simplicity, lets assume a simple circle with radius $R$ in 2 dimension. Therefore ...
0
votes
1answer
15 views

Distribution function of the random variable $R_2=e^{-R_1}$

An absolutely random variable $R_1$ is uniformly distributed betweem $-1$ and $+1$, find the density and the distribution function of the random variable $R_2$, where $R_2=e^{-R_1}$. $R_1$ is ...
2
votes
1answer
122 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
1
vote
1answer
25 views

Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
8
votes
3answers
213 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 ...
2
votes
1answer
34 views

Verify conditions of the Extension Theorem for a probability measure $\mathbb P$ of the Uniform[0,1] distribution

I am (self-)studying the book by Rosenthal called A first look at rigorous probability theory. My question is on verifying the conditions on a probability measure $\mathbb P$ of the Uniform[0,1] ...
-2
votes
1answer
41 views

A basic question on uniform distribution [closed]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
1
vote
1answer
44 views

Variance of a polynomial series from a uniform distribution

I intend to derive the variance of $Z$: $$Z \equiv \alpha_0+\alpha_1X+\alpha_2X^2+\dots + \alpha_MX^M = \sum_{m=0}^{M}\alpha_mX^m $$ for some $0 < M < \infty$ where each $\alpha_m \in ...
4
votes
2answers
112 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 ...
2
votes
2answers
41 views

Difference Uniform rv's

Let $U_{1}\sim U(0,1)$ be a standard uniform random variable. Is $U_{1}-U_{1}$ uniformly distributed? I've been trying to work this out as follows: Let $A,B$ be rv's $$P(A-B\leq ...
1
vote
1answer
37 views

Moment of uniform distribution

Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$. Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ ...
2
votes
2answers
105 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 ...
0
votes
2answers
154 views

Transformation of a uniform distribution in order to get a random variable distributed like Y.

$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$. is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known constant and ...
0
votes
2answers
73 views

Calculate expectation and variance

Let $(X_n)$ be a sequence of independent RVs which are uniformly distributed on $[0,1]$ interval. For $0<x\le 1$ we define $$N(x):=\inf\{n:X_1+\dots+X_n\ge x\}.$$ Show that $$\mathbb{P}(N(x)\ge ...
3
votes
1answer
172 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
votes
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 ...
2
votes
1answer
266 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
votes
1answer
346 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: ...
1
vote
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 ...
-2
votes
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 $P(x_2/x_3 \leq a)$?
0
votes
2answers
255 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
votes
0answers
679 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 ...
1
vote
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 ...
1
vote
2answers
175 views

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 ...
1
vote
2answers
276 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
837 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 ...
0
votes
1answer
309 views

Demonstrations of the expectation and variance of discrete uniform distribution

I'm studying for my exam of probability distributions and in my study book got these equalities: = Also mention that: is equal to how to reach these inequalities?