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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53 views

A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability B arrives before A?

As the title indicated, A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability that One day B arrives before A? I am bit confused whether to use Poisson (because arrival ...
-3
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2answers
41 views

If $X_1,X_2,X_3$ are three different uniform random variables, calculate $E(X_1 - 2X_2 + X_3)$. [closed]

Suppose that a random variable $X_1$ is distributed uniform $[0,1]$, $X_2$ is distributed uniform $[0,2]$ and $X_3$ is distributed uniform $[0,3]$. Assume that they are all independent. a) Calculate ...
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0answers
27 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 \...
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2answers
47 views

If the area of a square is uniformly distributed, then find the expectation and variance of the side length.

I'm told that the area of a square (denoted $A$) is uniformly distributed across the interval $[15,20]$, I'm then asked to find the expectation and variance of the side length but I can't work out how ...
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2answers
74 views

Probability that the inpatient guy arrived first

So the question asks: Tom and Jerry set up a meeting at a restaurant. Each one of them, independently of the other, arrives at some random time between 9:00 pm and 10:00 pm (that is, the arrival ...
3
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2answers
43 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let $P\in\{0,1\}^{n\...
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1answer
35 views

Combine discrete uniform distributions to achieve a discrete uniform distribution of a larger range?

How can I effectively combine multiple discrete uniform distributions of a limited range to achieve a discrete uniform distribution of a large range? I.e. given $unif\{a,b\}$ generate $unif\{a,c\}, c&...
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1answer
77 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
3
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1answer
34 views

Is modular multiplication under a prime modulus uniformly distributed?

Let's say that I have a prime, $p$, and an $m \in Z_p^*$. Then, I draw $a \leftarrow Z_p^*$ uniformly at random. Will $am \mod p$ be distributed uniformly over $Z_p^*$?
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0answers
35 views

Rewiting an integral in the case of a uniform distribution

I would like to rewrite this equation: \begin{equation} \widetilde{A}(\varphi^*) = \left[\frac{1}{1-G(\varphi^*)}\int_{\varphi^*}^{\infty} A(\varphi)^{1-\sigma}g(\varphi)d\varphi\right]^{1/(1-\sigma)} ...
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1answer
31 views

Uniformly Distributed Random Variable over an interval

Let $X$ be a random variable uniformly distributed over a nontrival interval $[c,d]$, and let $Y = aX+b$. For what choice of real constants $a$ and $b$ is $Y$ uniformly distributed over [0,1]? How ...
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0answers
27 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

The Statement of the Problem: Let $ \{ U_i \}$ be a set (sequence?) of iid random variables such that $U_i \sim \text{Uniform}(0,1)$, and define $$ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\...
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2answers
69 views

Probability that sum of independent uniform variables is less than 1

I would like to determine the probability $\mathbb{P}(X_1+\dots+X_n\leq 1)$, where $X=(X_i)_{1\leq i\leq n}$ is a family of independent uniform random variables on $[0,1]$. My first idea is to do this ...
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0answers
30 views

Probability function of the sum of multiple independent uniform distributions

Given n number of variables with uniform distribution. What is the probability distribution of the sum of these variables? Let's say that $a_1$ and $a_2$ are two independent uniform distributions in ...
0
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1answer
34 views

CDF and PDF of semaphore waiting time

Imagine we have a semaphore that alternates every 40 seconds between green and red. Waiting time is 0 when the semaphore is green, and when it is red it is the remaining time until it turns green. I ...
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2answers
60 views

What is the expected value of the sum of the $k$ (out of a set of $n$) smallest uniform random variables?

I know that the expected value of the sum of $n$ random variables is the sum of the expectation of each one. The expected value of a uniformly distributed random variable $U(a,b)$ is also well known ...
0
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1answer
23 views

subset of a uniform random number.

I need to create a 2-D random number generator that generates numbers in a convex region. Say, for example, this region falls within the area [-1,1]2. Would the following process lead to a uniform ...
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0answers
30 views

finding probability using uniform probability density

I am trying to solve: A value x is chosen with uniform probability density from the interval $[0, 1]$. Calculate the probability that $x < 1/2$ given that $x < 2/3$. I plotted these lines on ...
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3answers
55 views

pdf for random variable over unit disk

Let $(X,Y)$ be uniformly distributed on the unit disk $S = \{(x,y) \in \mathbb{R}^2:x^2 + y^2 \leq 1\}$. a) Find the probability density function for the RV $U = X + Y$. b) Find the probability ...
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1answer
23 views

CDF of the highest result of multiple unform random variables.

Say I have multiple uniform random variables. I want to know the CDF for selecting the highest result of all the variables. As an example, say I have 3 uniform random variables from [0, 100). Using a ...
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0answers
20 views

Distribution of maximum frequency in uniform sample

If I take $n$ random integers from $1$ to $m$, how do I calculate the distribution of the number of occurrences of the most frequent number? Any hints or initial approaches? I thought to get the ...
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1answer
46 views

Finding MLE of uniform distribution with actual example values

I'm watching this video and going to part I am stuck at here https://youtu.be/XaAtkCzdjLE?t=6m2s Following the example in the video, I assume that $\theta$ will be between $14$ and $501$. Now I don'...
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1answer
50 views

Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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0answers
2k views

Mean distance of random points on a rectangular grid

I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate ...
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3answers
64 views

Joint probability uniform distribution

I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one: We agree ...
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2answers
88 views

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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2answers
44 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
42 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
55 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
16 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: http://crypto.stackexchange.com/questions/32427/secure-blinding-factor-switching-at-malicious-server-...
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1answer
72 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
33 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
26 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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1answer
60 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 $T_x$...
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0answers
40 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
35 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 $N=\max(X_1,X_2,\dots,...
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3answers
54 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 $N=\max(...
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2answers
56 views

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

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
31 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 people,...
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1answer
49 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 example,...
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2answers
80 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 ...
0
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1answer
28 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} 1,...
1
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3answers
115 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 $P(X_1<X_2<X_3) +...
2
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2answers
50 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 ...
0
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2answers
56 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
35 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 ...
1
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1answer
36 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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0answers
18 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 $S={\...
1
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1answer
31 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 ...