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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Product of i.i.d. random variables uniformly distributed on $(-1,1)$ converges almost surely to $0$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.
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22 views

Find CDF of $Z:=\min\{X,Y\}$

Let be $X,Y$ independent random variables with $X\sim\mathcal{U}(\{0,1\})$ and $Y\sim\mathcal{U}(\{0,1,2\})$. Compute the CDF of $Z:=\min\{X,Y\}$. My idea: $$ \begin{align*} P(\min\{X,Y\}\leq x) ...
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1answer
33 views

Show that $P(X=2) = \frac{2}{3} - \frac{13}{27}$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U = p$ is Geometric with parameter $p$. Show that ...
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3answers
34 views

If $N_{\mid\Lambda = \lambda} \sim$ Poisson ($\lambda$) and $\Lambda \sim$ unif$(0,5)$, find the probability of zero occurring.

The number of storms in the upcoming rainy season is assumed to be Poisson distributed, but with a parameter $\Lambda$ that is also random and uniformly distributed on $(0,5)$. That is, $\Lambda \sim$ ...
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0answers
15 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
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1answer
53 views

Explain why $E(X)=1.65$ and $Var(X)=1.64$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U=p$ is Geometric with parameter $p$. (a) Find ...
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2answers
29 views

Probability that 4/4, 2/4 Discrete Uniform RVs aren't equal

If I have four discrete $uniform(1,32)$ RVs. I'm trying to figure out the probability that a) None are the same b) exactly two are the same I thought that the probability that none are the same ...
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0answers
20 views

Sum of two random variables - uniform distributions [duplicate]

I have two continuous uniform random variables I need to add. I read that to get the sum of two pdfs you convolve them. I'm getting a bit confused on the limits of integration though. If both RVs are ...
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2answers
51 views

Show that a random variable $T_x$ is uniformly distributed given that $T$ is uniformly distributed?

We have a lifetime $T$, which is uniformly distributed over $(0,b)$. We then introduce a new r.v., $T_x=T-x$, which is defined on $0<x<b$. We want to show that given $T>x$, the variable ...
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0answers
34 views

Joint density of Triangular RV and Maximum of Triangular RVs, parameterised by Uniform RV

Let $x$ be drawn from the uniform distribution on $[0,1]$. $x$ parameterises the Triangular distribution $Y$ with support $[0,1]$. I.e., $$ f_Y(y_i | X = x) = \begin{cases} \frac{2y_i}{x} \quad ...
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2answers
33 views

Find the first and second moments of a distribution of order statistics?

I'm not totally sure how to even word this question, but I need to find the first and second moments of two variables, $M$ and $N$ as defined by: $M=\min(X_1,X_2,\dots,X_n)$ and ...
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3answers
35 views

Find the joint distribution of the continuous order statistics?

Take $n$ independent variables, ${X_1, X_2,\dots, X_n}$, which are uniformly distributed over the interval $(0,1)$. Then, introduce the variable $M=\min(X_1, X_2,\dots, X_n)$ and the variable ...
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2answers
49 views

Compute $\mathbb{E}(U_{1} | M)$ and $\mathbb{P}(U_{1}=M)$

I am having trouble getting started on and finishing this problem. Any help that can be offered would be greatly appreciated. Let $U_{1},...,U_{n}$ be independent random variables uniformly ...
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1answer
20 views

Probability that a sample represents between X% and Y% of the population

Really no idea how to go about this. I thought about using a uniform normal distribution law but the answers I got made no sense. In a country that has a population between 1500000 and 3000000 ...
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1answer
44 views

RVs independently and uniformly distributed on interval $[0,1]$, prove every order is equally likely

Finitely many random variables $p_1,p_2,...,p_n$ are independently and uniformly distributed on interval $[0,1]$. They form an ascending sequence $p_{i_1} \le p_{i_2} \le ... \le p_{i_n}$. For ...
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2answers
45 views

The distribution of the x-coordinate on unit circle

I'm trying to determine the distribution of the x-coordinate (uniformly distributed) on the unit circle (density function). I've seen some threads on this, such as this, where they use the method of ...
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0answers
24 views

Could you please explain how do we get the second term while computing $E[f(Y)]$

I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform ...
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1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
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4answers
99 views

Find the probability of $X_2$ lying between $X_1$ and $X_3$

All $X_1,X_2,X_3$ are independent and uniformly distributed on $[0,1]$. Find the probability of $X_2$ lying between $X_1$ and $X_3$ Is the following method correct? Find the ...
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2answers
40 views

Probability of getting the same number three times.

If I have a set of numbers $\{1 \ldots n\}$ where $n \ge 1$ and I pick $3$ numbers from the set independently and uniformly. Whats the probability I'll get all $2$'s, the probability I get all the ...
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2answers
45 views

Find the probability of $\max[X,Y]<\frac{1}{2}$?

$X$ and $Y$ are uniformly distributed random variable on $[-1,1]$. Find the probability of $\max[X,Y]<\frac{1}{2}$. I calculated it using graphically that the region where the where $[X,Y]$ is ...
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0answers
19 views

Uniform distribution density function

Let's say we have two random variables $T_1$ and $T_2$ and the joint density function of the two is uniform over the region $0\leq t_1\leq t_2 \leq L$, where L is a positive constant. Then the area of ...
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1answer
30 views

question about the conditional disjoint probability.

I am trying to solve a problem like the following. Q) The event that a man arrives at a bank $\sim Poisson(\lambda)$. If two men visited the bank between 9:30 AM and 10:30 AM, what is the ...
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16 views

Find probability coverage under a discrete uniform

Let $Y_1=\theta+\epsilon_1$ and $Y_2=\theta+\epsilon_2$ where $\epsilon_1$ and $\epsilon_2$ are iid uniform in {-1,1}. Define the confidence set S for $\theta$ as: $S={Y_1-1}$ if $Y_1=Y_2$ and ...
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1answer
25 views

Math Behind Producing Uniform Distribution

I am familiar that if one can produce a uniform distribution, doing so, one can then produce random numbers for other types of distributions. I have tried reading some articles online but I am still ...
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1answer
65 views

All Cdfs have a uniform distribution on [0, 1]?

Consider the following proposition Proposition C Let $Z = F(X)$; where $F$ is the continuous cumulative distribution function of the random variable X, then $Z$ has a uniform distribution on ...
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1answer
47 views

Why is $X/\|X\|_2$ uniformly distributed on a unit sphere when X is n-dimensional standard gaussian vector?

In the proving the above, I see that since $X$ is multivariate gaussian then for any orthogonal matrix $Q$ we have that $QX$ is standard multivariate gaussian. Then I somehow reasoned that ...
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1answer
48 views

Uniform distribution over the unit disk

Suppose that $U_1$ and $U_2$ are independent, and identically and uniformly distributed over the unit disk, i.e., for $i = 1,2$, $U_i = (X_i, Y_i)$ and the joint density is \begin{equation} ...
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60 views

Pseudorandom Number Generator Using Uniform Random Variable

I am working out of Mathematical Statistics and Data Analysis by John Rice and ran into the following interesting problem I'm having trouble figuring out. Ch 2 (#65) How could random ...
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43 views

find limit distribution by using central limit theorem.

$$x_1,...,x_n \sim \text{uniform (0,1)}$$ $$Y_n=\sum_i^n X_i$$ I want to find limit distribution by using central limit theorem. $E(Y_n)=n/2$ and $V(Y_n)=n/12$ And Moment generating function ...
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1answer
32 views

Distribution of a Poisson process with uniformly random parameter

Let $X = Unif(2, 4)$ and $Y=Poisson(X)$. My goal is to find $P(Y=n)$, but I always seem to get stuck on some nasty integral. Here's what I've tried: $P(Y=n) =\int_2^4P(Y=n|X=x)*P(X=x)dx = \int_2^4 ...
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48 views

Probability problem: length of new segments

I have a line of length $l$. I divide the line in $n$ segments. I do this by choosing $n - 1$ random points (I mean that the $n - 1$ points are uniformly distributed from $0$ to $l$). I want to add a ...
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1answer
53 views

Two step random experiment: Density of combined uniform and normal distribution

imagine a random experiment, where first some number $u$ is drawn uniformly on $[c-\varepsilon,c+\varepsilon]$ for $c>0$ and $0<\varepsilon<c$. Next, a $N(u,\sigma^2)$-distributed random ...
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2answers
27 views

Sum of X-Uniform(0,1) + Y-Uniform(0,2)

I'm trying to find the CDF the sum of $X$ and $Y$ (which are independent). $X$ is uniform distributed over $[0,1]$ and $Y$ over $[0,2]$. I've seen some similar questions which explain the situations ...
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3answers
48 views

Generate prime of x decimal digits using bit-oriented prime generator

I've got a question on stackoverflow where somebody asks to generate a random 18-digit prime. Unfortunately, the only prime generator is the one from OpenSSL. This prime generator is however geared ...
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2answers
41 views

How do I renormalize these probability distributions?

So I have two random variables, $X_1$ and $X_2$, both uniformly distributed on $[0, 1]$ If $Y = (X_1 + X_2) / 2$, it will also be distributed between 0 and 1, but it won't be uniformly distributed ...
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1answer
35 views

Poisson $ (\lambda) $ process: Relation between $ n^{th} $ arrival time $ S_n $ and maximum of $ n $ arrival times

Poisson $ (\lambda) $ process: Relation between $ n^{th} $ arrival time $ S_n $ and maximum of $ n $ arrival times Sheldon Ross's Stochastic Processes, second ed: Consider a Poisson ($\lambda$) ...
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How to quantify the “uniformity” of a distribution of holes in a surface

I want to try to quantify if the distribution of holes over a surface is uniform or not. The holes can have any given shape and can be arranged in any way over the surface. Three examples are ...
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1answer
31 views

expectation and geometrical probability problem [closed]

A horizontal line of length $5$ units is divided into two parts. If the first part is of length $X$. Find $E[X]$ where $E[\cdot]$ stands for expectation. how to approach this question ? X can take any ...
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22 views

Finding conditional probability distribution (X|Y) from (Y|X)

$(Y,X)$ has a joint distribution where the marginal of $X$ is a standard normal and $Y|X \sim U \left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right] $ where $U[a,b] $ means uniform in the interval [a,b]. How ...
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Uniform distribution of points on a cone constrained to a continuous line

I was hanging lights on a Christmas tree yesterday, and thought of a problem , which may have an easy solution - but not one that I can think of off the top of my head. It is posed as follows: ...
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30 views

Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
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1answer
42 views

Detecting corrupted data in birthdates of a population

I have a population of N birthdates. Let's assume that birthdates are uniformly distributed over the year. I'm concerned that some of these records have been corrupted, for example by someone ...
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0answers
26 views

Cramer-Rao Uniform Distribution

If my data, $X_i\sim U(0,\theta)$ is iid. What is the Cramer-Roa lower bound for a variance estimator such as the sample variance? $ S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2$ I am stuck ...
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33 views

Expected value for number of occurences

Let $W$ be a random word made from letters which are in set $K$ (letters are uniformly distributed in $W$) . Suppose also that $W$ has finite length ($\geq 2$) and size of $K$ is finite ($\geq 2$). ...
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2answers
70 views

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, ...
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0answers
25 views

Expectations of order statistics of uniform RVs, via exponential formulation

If $U_1,\ldots, U_n$ are i.i.d. uniform random variables, then I know that the order statistics satisfy $$(U_{(1)},\ldots, U_{(n)}) \overset{d}{=} \left(\frac{X_1}{\sum_{i=1}^{n+1} X_i}, ...
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2answers
43 views

Probability of random variable with uniform distribution on an interval

Let a random variable X have a uniform distribution on the interval $[0, 10]$. Find $P(X(X + 10) > 11)$ Since X has a uniform distribution, the pdf of X is $$ ...
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1answer
15 views

Prove that two random variables are dependent

Given two random variables X and Y where X is uniformly distributed on [-1,1] and Y = X^2, prove that these two random variables are dependent. Of course, it's clear that they are dependent. But, how ...
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1answer
70 views

$X_1$, $X_2$ i.i.d RVs, $X_1$ is uniformly distributed. Show $E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$

Let $X_1$, $X_2$ be two i.i.d. random variables and $X_1$ is uniformly distributed (discrete) on the set $\{1,2,3\}.$ Show that: $$E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$$ Can someone give me ...