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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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 $iid$ $unif(0,1)$, ...
0
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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}, ...
0
votes
2answers
44 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 $$ ...
0
votes
1answer
16 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 ...
2
votes
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 ...
1
vote
2answers
37 views

Convergence weakly to exponential random variable for this r.v.

Can I get some insight of how to solve this problem? Let $X_1, X_2, X_3, ...$ be i.i.d. copies of uniform random variable on $[0, 1]$. Let $M_n = \text{min}_{1\leq i <j \leq n} |X_i - X_j|$. Show ...
0
votes
2answers
64 views

Uniform Random Variable: Correlation and Independence

Let X be a uniform random variable defined on the interval $(0,1)$. If $Y = 6X^2−6X+1$, compute the correlation of X and Y . Are X and Y independent? Are X and Y uncorrelated? So my work is. $F(X) ...
0
votes
3answers
30 views

Find cumulative distribution function of uniform distribution

Random variable X has uniform distribution on $[0,1] \cup [2,3]$. Find cdf of variable X. I mean i do not know how to treat this on such strange interval.
1
vote
1answer
51 views

Mutual information for a continuous uniform distribution

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
1
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1answer
63 views

IID Uniform probability problem [closed]

Three students independently attempt to solve a problem. Assume that the times taken by each student to solve the problem are iid according to U(0,30). Find the probability that the student who fi ...
1
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0answers
16 views

Covariance of uniform random and indicator function dependant on it

Define $I = \begin{cases} 1,& \text{if } X\leq a\\ 0,& \text{if } X\gt a \end{cases}$ $X$ is uniform on $[0,1]$. We want to compute $Cov(I,X)$ which involves $E[IX]$. $E[IX] = ...
0
votes
0answers
21 views

Find probability from Uniform Distribution [duplicate]

Two numbers are selected at random from the interval (0,1). If these numbers are iid and uniformly distributed, find the probability that the three line segments found by breaking interval into three ...
3
votes
1answer
64 views

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 ...
-1
votes
2answers
39 views

The distribution of the sum and difference of independent uniformly distributed variables

Suppose $X$ and $Y$ are independent uniformly distributed on the interval $[-a/2,a/2]$. What is the density function of $Z=X+Y$ and of $Z=X-Y$? I know that it will be the convolution of ...
1
vote
1answer
37 views

Uniform Probobility Distribution Word Problem, grocery store checkout and meeting a friend

The amount of time spent waiting in line at a grocery store express checkout varies from 5 minutes to 15 minutes and follows a uniform distribution. Let X be the amount of time spent waiting in line. ...
4
votes
0answers
83 views

$E[Y^2| (X-Y)^+]$ 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 ...
1
vote
0answers
35 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
0
votes
1answer
84 views

Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
1
vote
1answer
53 views

Find the pdf of $Y = g(X)$, where $X$ is a uniform random variable

The question is as follows: Let $X$ be a uniform random variable over $(-1,2)$. Let $g(x) = |x|$. Find the pdf of $Y = g(X)$. And here is my take so far: $$f(x) = \begin{cases} 1/2 & \text{ ...
1
vote
1answer
33 views

Transformation of variables for a non-monotonic function

Question: Let $U \sim \mathrm{Unif}(−α, α)$ follow the uniform distribution on the interval $(−α, α)$ for some parameter $α > 0$ and consider the transformed random variable $X = \sin(U)$. ...
1
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0answers
27 views

Distribution of function of uniformly random variables

I am sorry if there is no simple answer to this or the answer is completely obvious but I am approaching my wits end here. Probability isn't my forte, nor am I even a mathematician. I am essentially ...
0
votes
1answer
188 views

Uniform Distribution with Independent Random Variables to compute mean of the present value of a bond.

John wants to purchase a bond which will pay him $X$ thousand dollars after two years, where $X$ is equally likely to be any of the numbers in the set $\{0, 1, 2, 3, 4, 5\}$. John believes that the ...
0
votes
1answer
20 views

Uniform Distribution - Change of Variable

I have been stuck on the following question If $X$ has a cumulative distribution $F(x)$, then show $Y = F(X)$ has a uniform distribution with $U(0,1)$. I attempted to solve this problem by first ...
0
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0answers
13 views

Uniformly distributed arrival times, each willing to wait 15 minutes, what is the probability they meet? [duplicate]

Alice and Bob agree to meet for lunch on a certain day at noon. However, neither is known for punctuality. They both arrive independently at uniformly distributed times between noon and 1 pm on that ...
0
votes
0answers
36 views

Find the joint distribution of 2 random variables

Let $X$ and $Y$ be independent random variables with uniform distribution over $[0,1]$. Let $Z = XY$. Find the joint distribution function of $X$ and $Z$. So far what I have is this: $$P(X \leq t, XY ...
0
votes
3answers
38 views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
1
vote
1answer
25 views

When is an improper Riemann integral equal to Lebesgue integral

My original problem is given $X_i\sim^{iid}U[0,1]$, find $$\lim_{n \rightarrow \infty} (X_1X_2 \cdots X_n)^{1/n} = \lim_{n \rightarrow \infty} (\prod_{i=1}^{n} X_i)^{1/n}$$ Well, $$\lim_{n ...
0
votes
1answer
39 views

Uniform distribution on $[0,1]$ and random variable $Y=\frac{U}{e^{1-U}}$

$U$~$Unif[0,1]$ and we have the random variable $Y=\frac{U}{e^{1-U}}$. Find the density function of $Y$. So far I have that $0\le Y\le1$ and that... $$F_Y(t)=P(\frac{U}{e^{1-U}}\le t)=P(\ln ...
0
votes
0answers
104 views

p-value of uniformity of given distributions,Matlab

Given a vector of real numbers $[a_0,...,a_n]$, how do I find the $p$-value (in Matlab, say) that it is drawn from the uniform distribution over [0,1]? I.e. $H_0$ is the hypotheses that it is drawn ...
1
vote
2answers
59 views

Probability Xavier and Yolanda meet for lunch

Xavier and Yolanda plan to meet for lunch between noon and 1 p.m. They arrive independently with uniform distribution on [0, 1]. Yolanda will wait 30 min. for Xavier, but Xavier will only wait 15 min. ...
0
votes
1answer
28 views

$P\left(X+\frac{10}{X}>7\right)$ of a uniform distribution

Problem: $X$ has a continuous uniform distribution on $[0,10]$. Find $P\left( X + \frac{ 10 } { X } >7\right)$. So far, I have the PDF $f(x) = 1/10$ and CDF $F(x) = x/10$ for $0 < x < 10$. ...
1
vote
1answer
35 views

Density function of uniform prob distribution

Let $X ∼\operatorname{Uniform}(0,1)$. Find the density function of $Y = e^X$. I got to: $F_Y(y)$=$P(Y\le y)$=$P(e^X\le y)$=$P(X\le \ln(y))$ Not sure where to go from here?
2
votes
1answer
31 views

Uniform distribution probability calculation

Here is an exam problem with the work shown: A man and a woman agree to meet at a certain location at about 12:30 pm. The man will arrive at a time uniformly distributed between 12:15 and 12:45, ...
1
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0answers
10 views

Partition Theorem to show $P(W \gt Z)$

I am confused as to how to use the partition theorem on the following example? Any help is appreciated! Suppose that W has a U(0,1) distribution and suppose that W is independent of the random ...
0
votes
0answers
36 views

approximate a probability distribution by moment matching

I have a 60-40 weighted distribution, of uniform(0,7.5) and uniform(7.5,10) respectively, i.e. $$f_X(x)=(0.6/7.5)1_{x∈[0,7.5)}+(0.4/2.5)1_{x∈[7.5,1]}$$ I have worked out that $$E(X) = 0.6(7.5/2) + ...
0
votes
1answer
48 views

Variance of a weighted uniform distribution

Given a weighted uniform distribution, where it is a 60-40 mixture of uniform(0,7.5) and uniform(7.5,10), I have found the mean to be $$E(X) = 0.6(7.5/2) + 0.4((10+7.5)/2)$$ How do I find the ...
1
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1answer
27 views

Multivariate transformation of three independent variables

An insurance company offers the following insurance package to a customer with three businesses on the same street. The insurer will pay for all damages to the business that incurs the most damage, ...
0
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0answers
21 views

Identify a probability distribution with coordinate transformation

I have a problem with this task: We have a random variable $X:\Omega \rightarrow \mathbb R^2$, which is uniformly distributed on $K:= \{(x_1,x_2) \in \mathbb R^2 : \sqrt{x_1^2+x_2^2} \le 1 \}$ Now I ...
0
votes
0answers
40 views

Uniform distribution

Let $X_{1} ..., X_{n}$ be a sample from $U([0,\theta])$ for $\theta>0$, $X_{n:n}$ denotes the maximum observation, $\bar{X}=\frac{1}{n}\sum_{0 \leq k \leq n}X_{k:n} $ compute probability $$ ...
0
votes
2answers
56 views

PDF of several draws from an uniform distribution?

Suppose I draw several times from an uniform distribution, $X\sim\mathcal{U}(0, 1]$. (I'll use $\mathrm{R}()$ to denote an independent drawing.) What is then the PDF of several draws, added and/or ...
0
votes
1answer
21 views

$X$ RV with cdf $F$, $W \sim U[0,1]$ independent $\Rightarrow$ $V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1]$

I try to prove: Let $X$ be a discrete random variable with cdf $F$, $F_{-}(x):=P(X<x)$, $W \sim U[0,1]$ a random variable and $X, W$ independent. Then $$V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1].$$ My ...
0
votes
0answers
12 views

Mapping Function for Non-Uniform Circular Distributions

Note: I a programmer, not a mathematician. For a non-uniform distribution of points that occur on the unit circle is there a function or mapping that returns the k$th$ point in this distribution? For ...
0
votes
0answers
35 views

Uniform distribution over $\mathbb{R}^2$

Suppose, on $\mathbb{R}^2$, that $X$ is a random variable which takes values uniformly at random over the $\textit{line segment}$ from $(0,0)$ to $(a,a)$, where $a > 0$ is a positive constant. How ...
0
votes
0answers
31 views

poisson and uniform distributions

I have an answer to this question from someone else but I do not think it is right. Here is the question: Customers arrive at a bank at a Poisson rate lambda. Suppose two customers arrive during the ...
1
vote
1answer
31 views

Compound of uniform and gamma probability distributions

I am trying to compute the distribution of a uniform distribution whose upper limit is drawn from a gamma distribution. That is, $X \sim \Gamma(\alpha,\beta)$ $Y \sim U(0,X)$ We know: ...
1
vote
1answer
26 views

MLE of uniform distibution again

I've struggled for hours with a seemingly simple problem, I'm supposed to compute the MLE for $\theta$. We have $(y_1, y_2...y_n)$ obervations with a uniform distribution. The density function is as ...
0
votes
1answer
24 views

Probability of A winning given a uniform distribution

If the interval of A has been uniformly chosen as [0,1] and B as [0,6] then what is the probability of A being a lower number than B? I'm completely lost here, do I somehow calculate the uniform ...
0
votes
0answers
14 views

How do I normalize a uniform dist?

If I have a uniform distribution over A to B, and I want to find the prb of a trial being within 1 std dev, once I have the mean and std dev, how do I normalize this, so the mean is 0 and a std dev is ...
1
vote
1answer
36 views

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 ...
1
vote
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?