# Questions tagged [uniform-distribution]

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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### Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$

Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$ where $X_{(n)} = \max_i X_i$. It was suggested to use Basu's Theorem which I am ...
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### Probability that the bus will arrive within 4-5 minutes

The bus runs at intervals of 10 minutes, and at a random moment you come to a stop. What is the probability that the bus will arrive within 4-5 minutes? My textbook says the answer is $\frac{1}{10}$ ...
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### moment generating function uniform distribution

I'm trying to prove a simple example of MGF for $E[x]$ for a simple RV with uniform distribution over range $[0,10]$ but not sure where I'm getting stuck: $A(t) = E[e^{tx}] = \int_{0,10}e^{tx}dx/10$ *...
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### Entropy of discrete and continuous uniform distributions

Despite a similar post here, I read that the entropy of a uniformly distributed discrete random variable is always log base $2$ of the number of observations in the dataset, $H(X) = \log(N)$. Is this ...
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### Suppose $U \sim Unif(0,1)$ and $Z \sim Unif(U,3+U)$. How can I find the pdf for $U + Z$?

Suppose $U \sim Unif(0,1)$ and $Z\mid U \sim Unif(U,3+U)$. I would like to find the pdf for $U + Z$, which in my process on $Z$ is a continuation of $U$. Is there a straightforward way to derive this?
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### Finding the joint distribution of two independent uniform random variables $X,Y$ given event $E$

If $E = \{\text{either} \ X < 1/3 \ \text{or} \ Y<1/3\}$ for $X\sim Unif(0,1)$ and $Y\sim Unif(0,2)$, does the joint distribution of $X,Y$ given event $E$ exist? I am assuming that $X$ and $Y$ ...
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### What is the pdf for a jointly uniform distribution inside a triangle?

I have a triangle bounded by $0 \leq x, y \leq 1$ and $x + y \geq 1.5$. I'm told that points are uniformly distributed within this triangle. I am wondering how I can find the pdf? Is it simply solving ...
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### Probability two uniform distribution(0,1) = 2/9

Two numbers are independently and uniformly chosen from the interval (0,1). What is the probability that the sum of the numbers is less than 1 and the product of the numbers is less than 2/9? (Note ...
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### What is the pdf of $\frac{|x-y|}{(x+y)(2-x-y)}$ when $x,y$ are i.i.d uniform on $[0,1]$?

If $x,y$ are i.i.d uniform random variables on $[0,1]$. I know that the PDF of $|x-y|$ is: $$f(z) = \begin{cases} 2(1-z) & \text{for 0 < z < 1} \\ 0 & \text{otherwise.} \end{cases}$$ I ...
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### Uniform distribution and conditional distribution

If $U \sim Uni([0,1])$, does $U| U\ge 0.5 \sim Uni([0.5,1])?$ If it's false, why? and if it's true, can it be generalized for any Uniform Distribution?
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### If $U$ is uniformly distributed on $S^{d-1} \subset \mathbb{R}^d$, what's the distribution of its orthogonal projection onto any vector?

Let $U \in S^{d-1} \subset \mathbb{R}^d$ follow a uniform distribution on a sphere. Let $v \in \mathbb{R}^d.$ Then is the orthogonal projection $U^{T}v=\langle U,v \rangle$ uniformly distributed, and ...
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### Find pdf of transformation of two random variables using CDF [duplicate]

Let $X,Y \sim$ Uniform$(0,1)$ be independent. Find the PDF for $X/Y$. Let $Z=X/Y$. We want to find $F_z(z)=P(Z \leq z)=P(X/Y \leq z)$. We can make $Y$ super small with fixed $X$, and conversely we ...
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### Probability that $\max(X_1, \ldots, X_n) - \min(X_1, \ldots, X_n) \leq 0.5$

Consider $n$ IID random variables $X_1, \ldots, X_n \sim U(0,1)$. What is the probability that $\max(X_1, \ldots, X_n) - \min(X_1, \ldots, X_n) \leq 0.5$. Denote $Z_1, Z_n$ as the min and max ...
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### arctan of ratio of two normal variables is uniform

Say $X, Y$ are independent standard normals, and $\theta = \arctan(Y/X)$. Prove that $\theta$ is uniformly distributed over it's range. It is pretty intuitive that the distribution of $\theta$ would ...
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### $(X,Y)$ on a triangle

Let $(X,Y)$ a random variable uniformly distributed on the triangle $(0,0)$, $(0,1)$, $(1,0)$. Find the density of $(X,Y)$. $\rightarrow f_{X,Y}(x,y)=2$ Determine if $X$ and $Y$ are independent or ...
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### Probability question - Normal and Uniform

In a factory there are $2$ machines that create tubes (They are independent from each other) Length of the tubes of machine A is distributed normally with an expectancy of $101$ cm. and a Variance of ...
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