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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Meaning of probability density function - continuous random variables

Suppose we have a random variable X uniformly distributed over the interval (0,1). The probability density function of X is given by: $$f(x)=\left\{\begin{array}{l} 1 \space\space if \space\space ...
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1answer
23 views

Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?

How to show that $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent, if $X_1$ and $X_2$ are i.i.d. exponential with mean $1$? Is there a simple way to see this?
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36 views

Uniform Distribution Characteristic Function

What does it say about the uniform distribution that when generating the $n^{th}$ moments by the characteristic function we don't end up with cancellation of the imaginary values in the denominator ...
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2answers
46 views

Uniform Probability Distribution CDF and Probability

Suppose a value $x$ is chosen at random in the interval $[0,10]$. In other words, $x$ is an observed value of a random variable $X \sim \mathrm{UNIF}(0,10)$. The value $x$ divides the interval ...
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1answer
51 views

Prove the median of a uniform distributions is $\frac{1}{2}(a+b)$

Let X~U(a,b) with a and b in the real line, such that b>a with X's pdf given by $$f_X(x)=\frac{1}{b-a}\mathbb{1}(a<x< b)$$ Show the median of X's distribution is given by $$m=\frac{1}{2}(b+a)$$ ...
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33 views

Expectation of Uniform Distribution with Sin

Let $X ∼ \operatorname{Unif} (a, b)$. What is $E[\sin(X)]$? I know how to find the expectation of a uniform distribution, but I'm unsure how to find $E[\sin(x)]$. ...
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62 views

Height of uniform distribution?

If X follows a uniform distribution in the interval [2, 7], what is the height of the probability density function (pdf) at x = 4? I'm new to Probability & Statistics and will appreciate any ...
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43 views

Find the distribution of $|X-Y|$ if $X$ and $Y$ are i.i.d. uniform on $[0,1]$

$X$ and $Y$ are independent random variables uniformly distributed over $[0,1]$. I want to find the CDF of $|X-Y|$. I could use convolution but I wan't to calculate this more "directly". Here is my ...
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58 views

Construct random variable from uniform distribution [closed]

I am trying to do this problem: Suppose $ U$ is a random variable with distribution $\mathcal U(0,1)$. Find a function $g$ such that $g(U)$ has distribution: i)$\mathcal E(1)$ ii) ...
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1answer
17 views

Average distance to the closest neighbor

Suppose that 3-dimensional space contains countably infinite number of randomly positioned particles (points), on average $1$ particle per a unit of volume. Their distribution is homogeneous and ...
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2answers
92 views

Probability of waiting before meeting in case of two uniformly distributed random variables

I have a question that goes like this. Two people, X and Y, decide to meet at a particular time. The probability of them both being late is uniformly distributed between 0 and 60. Person X is always ...
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1answer
36 views

How can I find a minimal sufficient statistic for θ from a U(θ-1, θ+1)?

Suppose X1:n is a random sample from a U(θ-1, θ+1) distribution. Find a minimal sufficient statistic for θ. Show that the MLE of θ is not well defined. Suggest an alternative “sensible” point ...
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36 views

convergence in distribution

Let $U_{t}$ be iid Uniformly distributed on (0,1). Suppose $\hat{\theta}_{T}\stackrel{d}\rightarrow \theta^{*}$ with $\theta^{*}$ some random variable on (0,1). I believe $\sum_{t=1}^{T}I(U_{t}\leq ...
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29 views

Prove that uniform distribution on a set of vertices $V$ is stationary if the graph is regular.

I was going through Random walks on graphs: A survey It was stated that: Uniform distribution on a set of vertices $V$ is stationary if the graph is regular. Can anyone give me some hints to ...
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1answer
37 views

PDF for $\frac1a Uniform(−a,a)$

Problem: Let $X$ be $Uniform(−a,a)$ distrubuted. Calculate the PDF for $Z = \frac1a abs(X)$. Attempt: I think graphically here. $X$ is $U(-a,a)$ with $PDF = \frac1{2a}$ so $abs(X)$ is $U(0,a)$ with ...
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49 views

Statistics find E(x), the number of distinct elements in uniformly distributed pool of items

Question: Suppose there are Y types of balls in a bucket, which are normally distributed and independent. Hence the probability of picking one type out is $\frac{1}{Y}$. Let $x$ be the number of ...
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1answer
71 views

Discrete conditional probability and expectation

I'm having troubles in solving this probabilty problem. A group of $n$ players is given; they are divided into two teams with the following procedure. A number $X$ is chosen randomly from the set ...
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2answers
23 views

How can we find the distribution function of an Uniform Random variable with Random variable bounds?

X is a uniform random variable in (0,1) and Y is a uniform random variable in (X,1). How can I find the probability density function of Y? I thought and searched a lot and I found nothing. please help ...
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42 views

Expected value of a biased coin

I came across this question of finding expected value of p given Head showed up in last toss of coin. Here p is probability of a biased coin showing H and p is given as uniformly distributed.
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1answer
30 views

Calculation of quantiles of a uniform distribution over a sphere

How do we calculate quantiles of a uniform distribution over a sphere ? Can anyone provide me with a tutorial ?
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53 views

CDF of Function of a Random Variable

This question is pretty basic but I am still having trouble understanding it. This is the question, verbatim, from my textbook: "If $X$ is a random variable that is uniformly distributed between ...
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1answer
45 views

Variance of unbiased estimator of a random sample from the uniform distribution?

Let X1,...,Xn be a random sample from the uniform distribution on the interval from 0 to theta for some theta>0. I want to find the variance of the unbiased estimator. I know the unbiased estimator ...
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1answer
63 views

Uniformly distributed variables: what does the sum reveal.

Say $U_1, U_2 \sim U(1,0)$ are independent uniformly randomly distributed variables on $[0, 1]$. What lower bound $C$ should I enforce on their sum in order to believe (with a probability $p$) that ...
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27 views

C.D.F and measurablity

If $$F(x)=P[X \leq x]$$ is continuous in $x$, show that $Y=F(X)$ is measurable and that $Y$ has a uniform distribution $$P[Y\leq y]=y, 0 \leq y \leq 1$$ Proof so far: To show that $Y$ has a uniform ...
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62 views

Given 2 random points from a uniform distribution, what is this probability?

This question is very similar to (and is, in fact, an intermediate question of) this question. Therefore, much of this question is going to be very similar to that question. Suppose we're given 2 ...
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1answer
31 views

Jack the rabbit and joint distribution probabilities

I've ran across this problem in my textbook, really don't know how to solve it. Answer: .3973 What I've tried: Since the rabbit's location is uniformly distributed, and depends on both an x and ...
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1answer
175 views

Given 3 random points, what is the probability of these two situations involving a perpendicular bisector and distances?

Suppose we're given 3 random points $p_0=(x_0,y_0),p_1=(x_1,y_1),p_2=(x_2,y_2)$ from a two-dimensional continuous uniform distribution $\{U(a,b)\}^2$, for some $(a\in\mathbb{R})\lt (b\in\mathbb{R})$, ...
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39 views

What are the odds that one value picked from a uniform distribution is greater than any other?

Given the continuous uniform distribution $U(a,b)$ and any 2 values $u_1,u_2$ picked from $U$, what are the odds that $u_1 > u_2$?
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1answer
37 views

proving uniform distribution, random variables

I want to prove that if X is a random variable with the uniform distribution over [L, R] and Y = cX + d with c > 0, then the uniform distribution of Y is over the interval [cL + d, cR + d]. I'm using ...
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47 views

Density of sum of normal and cosine of a uniform random variables

Let X be a normal random variable with mean 0 and variance $\sigma^2$, let $\Theta$ be uniform on $(0,\pi)$, and let a be a real number. Assume X and $\Theta$ are independent. Find the density of ...
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91 views

The One-way Highway

This is supposedly a thought-provoking interview question asked, and I though I have an idea of a possible solution, I can't prove it. The question is the following: You have $n$ cars that are all ...
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1answer
122 views

How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
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1answer
44 views

Covariance of binomial variables

I can't demonstrate that the covariance of G is like in (3): I can't understand because there's that factor "n". Can someone help me please? Thank you!
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28 views

Uniformly distributed points in a spheroid

I want to pick points inside a spheroid with uniform probability. The problem was solved for a sphere here: Uniformly distributed points on a sphere. I know, that the "shooting method" works: Generate ...
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1answer
162 views

transformation of uniformly distributed random variable f(x)=1/2pi into Y=cosx

Let $X$ be a uniformly distributed function over $[-\pi􀀀;\pi]$. That is $ f(x)=\left\{\begin{matrix} \frac{1}{2 \pi} & -\pi\leq x\leq \pi \\ 0 & otherwise \end{matrix}\right.\\ $ Find ...
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3answers
71 views

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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Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
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14 views

Monotonicity of this conditional probabilty

$U_i ( i=1,...,n)$ are i.i.d and (0,1) uniformly distributed variables. Define its truncated product at threshold $\tau$ $(\tau\in(0,1))$ as $U=\Pi_{i=1}^nU_i^{I(U_i< \tau)}$. Define ...
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1answer
41 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
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45 views

uniform distribution, probability

This is regarding the previously asked question: Let $T_1$ and $T_2$ be random times for a company to complete two steps in a certain process. Say $T_1$ and $T_2$ are measured in days and they have ...
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0answers
61 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
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1answer
27 views

Explanation for “jointly pdf is constant but marginal pdf is not”

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$ From short computation, we know: Jointly pdf: $f_{XY}(x,y) = 2$ Marginal pdf of $x$: $f_{X}(x) =\int_0^x ...
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102 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: ...
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1answer
71 views

Expected value of sum of uniformly distributed variables

Let $X_i$ be a uniformly distributed random variable on the interval $[-0.5, 0.5]$ that is: $X_i$ ~ $U(-0.5, 0.5)$, for $i \in [1, 1500]$ How can I calculate the expected value of the sum ...
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3answers
125 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
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1answer
32 views

Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$. I know that maybe by using Characteristic function of $S_n$ ...
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2answers
66 views

Probability Uniform Distribution Set Up Integral

Consider a $1$ meter stick and suppose you break it into two pieces $X$ meters from the end, where $X \sim \operatorname{Unif}(0,1)$. What is the expected length of the longer piece (after it is ...
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116 views

Probability and Uniform distribution lottery question

Suppose that a person has a lottery ticket from which she will win $X$ dollars, where $X \sim\mathrm{ Unif} (0,4)$. Suppose her utility function is $U(x) = x\alpha$ for $x \geq 0$ and $0$ otherwise, ...
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2answers
52 views

Uniform distribution and expectation

Let $U \sim \mathrm{Unif}(0,1)$, $X=U^2$ and $Y=e^X$. Compute $E[Y]$ (leave answer as an integral). So essentially we need to compute $E[e^{U^2}]$? I am a little confused how to approach this problem? ...
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0answers
30 views

Departure from uniformity in a continuous (time) distribution

I know how to quantify the departure from uniformity ( or a uniform distribution) for discrete distributions. Assume you have a distribution set of P: ...