# Tagged Questions

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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### 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!
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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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### Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$\max(x_i) | \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b)$$ I agree that the probability ...
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### 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 ...
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### Sum of discrete and continuos random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are ...
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### Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
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### 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?
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### 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 ...
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### 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 http://...
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### $P\{B^{2}-4AC\geq 0\}$ 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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### 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 ...
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### 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]. ...
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### 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 ...
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### 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 ...
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### 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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### $E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$

I am preparing for a test and am unsure of my workings on this exercise: Calculate $E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$ Where $A^+ = \max (A,0)$ My approach was to calculate ...
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### 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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### Inductively defined random variables

Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$. Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the ...
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### 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?
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### 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 ...
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### Show that there is no discrete uniform distribution on N.

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?
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### 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 ...
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### A single, good test for a random number generator?

I'm chasing a bug in the RNG for a well-known programming language under certain pathological inputs. There is an obvious pattern in this pathological case, apparent with very small n (~ 10000), and ...
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### 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 ...
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### Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn't find the mean and variance of the order statistics of a discrete ...
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### 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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### 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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### Collisions with four bullets

This is a follow-up question of Colliding Bullets. I'm interested in a rigorous calculation of a specific aspect of the referred question. We consider four bullets. Once per second a bullet is fired ...
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### Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case 1: ...
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### 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 ...
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