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.
2,433
questions
-1
votes
0
answers
21
views
can a linear combination of normally distributed random variables be uniformly distributed? [closed]
With X ~ N(2,4), Y ~ N(5,7) and Z ~ N(-3,9) and Cov(X,Z) = -5, Cov(X,Y) = Cov(Y,Z) = 0. Given that 2X-4Y+9Z gets you a mean (-43) and variance (677). Is U[-43, 677] a valid linear combination for X, Y ...
- 1
0
votes
0
answers
58
views
Uniform distribution always exists on a closed convex compact set?
Let $X$ be a topological space with a weak order $\succsim$.
$C\subset X$ is a closed convex compact set.
What additional topological assumptions do we need to ensure that $C$ always has uniform ...
- 718
0
votes
1
answer
86
views
Recursive Uniform Distribution Expectation Question
Suppose we draw some k ~ Unif(0, 1). Then, we will draw some $u_1$ ~ Unif(0, 1). If $u_1 < k,$ we stop. Else, we will draw $u_2$ ~ Unif(0, $u_1$). We will continue drawing until $u_n < k.$ What ...
1
vote
1
answer
30
views
Expectation of sum of indicator function on uniform random variable
Let C be an estimator defined as
$$C = \frac{1}{n}\sum_{k=0}^{n}1_{U_k>a}$$
where $a$ is a real number in $(0,1)$ , $1$ is an indicator function and $U_k$ are uniform random variable in $(0,1)$.
I ...
- 929
1
vote
0
answers
10
views
Limit of expectation of reciprocal mean of uniforms
Suppose we have a sequence of iid unif$(0,1)$ random variables. I want to know whether or not the sequence of $\mathbb E[\frac1{\bar X_n}]$ converge to 2 (since $\bar X_n$ strongly converges to $\...
- 14.7k
1
vote
1
answer
67
views
Sum of uniform and beta distribution
Suppose $X ∼ Beta(a = 3; b = 1; θ = 1)$ and $Y ∼ U (−2, 2)$ are independent. Derive an expression for the cumulative distribution function of $X + Y$.
I am trying to do this by a convolution but I am ...
- 13
1
vote
1
answer
76
views
Question about the area element of picking points uniformly on a 3 dimensional sphere
Given the unit sphere in $\mathbb{R}^3$, we want to pick points uniformly across its surface. My first thought was to parameterize by spherical coordinates
$$
(\theta,\phi) \mapsto (\cos\theta\sin \...
- 247
1
vote
1
answer
48
views
Determine the probability mass function of $ Z=X+Y $
A random variable $ X $ is called (discretely) uniformly distributed on $ \{1, \ldots, n\}, n \in \mathbb{N} $ if it holds for its probability mass function (also called count density) $ \left\{p_{j}\...
- 606
0
votes
1
answer
62
views
The probability value of the three matrices
Determine the probability value that the following three matrices have real eigenvalues. For example, if the random variables $A$ and $B$ are given by a uniform distribution $(0,1)$, with $A$ and $B$ ...
0
votes
0
answers
65
views
Iterative Expectation of a Random Sequence a fucntion of Continuous Uniform Random Variable (Solved)
Problem: Let $X$ be uniformly distributed in the unit interval $[0,1]$, let
\begin{equation*}
Y_n=
\begin{cases}
0, & \text{if}\:X \geq \dfrac{1}{n} \\ \\
n, & \text{if}\:...
2
votes
1
answer
28
views
$X_i\sim \mathrm{UNIF}(0,\theta)$. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion.
Consider a random sample from a uniform distribution $X_i\sim \mathrm{UNIF}(0,\theta)$, where $\theta$ is unknown. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion.
...
- 65
1
vote
3
answers
102
views
If $X\sim \text{Uniform}[-1,1]$ and $Y = X^2$, what is $E[X\mid Y]$?
Given that $X$ is $\text{Uniform}(-1,1)$ continuous, $Y = X^2$, what is the expected value of $X$ given $Y$? I was asked to calculate the MMSE of $X$ given observing $Y$, which is $E[X\mid Y]$. The ...
- 13
0
votes
1
answer
58
views
This is a probability question once asked in gate exam can anyone please solve this [closed]
Subway trains on a certain line runs every half hour between midnight and six in the morning. What is the probability that the men entering the station at random time during the period will have to ...
- 11
1
vote
2
answers
44
views
Uniform Sampling points on a line using 2 Uniform distributions
I have been struggling with this problem for quite some time but I am not sure how to proceed.
So I am given a sampling algorithm : We would like to uniformly sample points on a line between A and B. ...
- 13
1
vote
0
answers
91
views
uniform random variables and their expectations
Let X and Y be independent uniform random variables on (0,1). Let $Z=\lfloor{\frac{1}{X+Y}}\rfloor$
I want to find $P(Z=0)$ and $E[Z]$
Firstly, I got that:
$$f_{X+Y}(a)=\begin{cases}
a&0\leq a\...
- 45
0
votes
0
answers
20
views
Elementary question about uniformly distributed random variable on specific set
Suppose that we have a set $\Omega\times [0,1]$ where $\Omega$ is finite and stands for the state space, and $X$ is a random variable that is indepedent of $\Omega$ and uniformly distributed on $[0,1]$...
- 145
2
votes
1
answer
55
views
$P(U_2<U_3<...<U_n$ and $U_1 \in (U_2, U_2 + c))$, $U_i$ are i.i.d uniform(0,1) r.v's and $c>0$
I'm trying to calculate the probability $P(U_2<U_3<...<U_n, ~\text{and}~~ U_1 \in (U_2, U_2 + c))$, $U_i$ are i.i.d uniform(0,1) r.v's and $a>0$ and $P(U_1<U_3<...<U_n, ~\text{and}...
- 31
7
votes
2
answers
335
views
Conditional distribution of sum of squared iid uniforms
Is it true, that if $U_{1}$ and $U_{2}$ are iid uniform distributed variables on $\left[-1,1\right]$, then the sum of $U_{1}^{2}$ and $U_{2}^{2}$ is still uniform distributed conditioned on the set ...
- 1,258
0
votes
0
answers
21
views
Question regarding bounds for Uniform Joint Distributions
I am doing joint probability and i came across a problem with bounds of the joint probability.
Let $X, Y, Z$ be independent and uniformly distributed over $(0, 1)$. Compute $Pr(X ≥
Y Z)$.
From the ...
- 35
0
votes
1
answer
45
views
Variance of sum of n distinct Uniform Random variables
I came across a problem in Quantguide which read:
Suppose you continually randomly sample nested intervals from
[0,1], halving the size each time. That is, the next interval is
[x,x+0.5], where
x∼U(0,...
0
votes
1
answer
54
views
lower bound on the squared sum of random variables
Let $X_1, \cdots, X_n$ be $n$ iid random variables such that each $X_i$ follows from the continuous uniform distribution in $[-1,1]$.
I am looking for a lower bound on
$$
\mathbb{E}\left[\sqrt{\sum_{i=...
- 638
1
vote
1
answer
44
views
Relation between outcomes of two random variables.
Even with the new Wolfram plugin for chatGPT, I'm struggling with the following question.
Suppose $X, Y \sim U(0, 2)$, i.e. both of them are independent and uniformly distributed. What would then be ...
- 13
0
votes
1
answer
69
views
How to derive a negative binomial distribution from an uniform distribution?
I want to generate a negative binomial distribution from an uniform distribution. My attempt:
U follows $Unif(0,1)$.$X =\lfloor ln(U) \rfloor$.
$P(X=x)=P(\lfloor ln(U) \rfloor=x)$
=$P(x<=\lfloor ln(...
- 1
0
votes
0
answers
38
views
What is the distribution of $Z = \left| X -Y \right|$ where $X$ is uniform from 0 to 2 and $Y$ is the nearest integer to X?
Here is my attempt so far, but by checking my equations numerically, my solution is wrong.
Since $Z = \left| X -Y \right|$, we can start calculating is distribution directly:
$$
\text{P}(Z \lt z) \: = ...
- 125
5
votes
2
answers
324
views
Find the Expected Travel Distance of the Ambulance (Where did I go wrong?)
Problem: The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is ...
- 143
2
votes
2
answers
114
views
Convolution of two uniform probability densities (two square waves)
Assuming $X$ and $Y$ are i.i.d. random variables let $Z = X + Y$
$$
f_X(x) = f_Y(y) =
\begin{cases}
1/2 & -1 \le x \le 1 \\
0 & \text{else}
\end{cases}
$$
Find the density ...
- 53
0
votes
0
answers
27
views
Average side length of a triangle with perimeter $p$
On the one hand, I think that by symmetry the average side of a triangle with given perimeter $p$ is $\frac{p}{3}$.
However (and here I'm probably mistaken), if I look at a side of the triangle, say $...
- 995
1
vote
1
answer
123
views
Expectation of a sum of uniform IID variables given the sum is greater than 1
Let $X_1, X_2$, ⋯ ∼ Unif(0,1) IID and $N_1$ = min{ n : $X_1 + ⋯ + X_n$ > 1}. Let $S_n = X_1 + ⋯ + X_n$. Compute E[$S_{N_1}$].
My approach:
$$
E[S_{N_1}] = E[E[S_{N_1}|N_1]]
$$
$$
E[S_{N_1}|N_1] = \...
- 45
0
votes
1
answer
29
views
Minimum number of draws in a Dirichlet distribution to obtain values above a threshold
I am generating a flat Dirichlet distribution such that $\alpha_{1},....,\alpha_{k} = 1$. The distribution looks even for values of $ k < 5$ and 1.000.000 draws. However when I increase the value ...
- 13
2
votes
1
answer
27
views
Help with starting a proof regarding empirical distribution function of a uniform distribution
Suppose $U_1,...,U_n$ is a simple sampling from a uniform distribution $U(0,1)$ and $G_n(u)$ is an empirical distribution function. Prove that
$$
\begin{gathered}
n \int_0^1\left(G_n(s)-s\right)^2 d s=...
- 23
0
votes
0
answers
31
views
Probability of having a disease given that 20 people have it out of 100 random people
Each person has a disease with probability $p$ independently. Out of $100$ random persons, $20$ tested positive. what is the value of $p$? Assume that the disease can be tested accurately with no ...
- 6,459
2
votes
0
answers
36
views
Division of tranformations of Uniform
Let $U_1 \sim U(0,1)$, $U_2 \sim U(0,1)$, $X_1 = U_1^{1/\alpha}$ and $X_2 = U_2^{1/\beta}$
$U_1$ is independent of $U_2$ and $X_1 +X_2 \leq 1$
Show that: $$\dfrac{X_1}{X_1 + X_2} \sim Beta(\alpha, \...
- 57
0
votes
0
answers
63
views
To find the expectation of a function of a uniformly distributed random variable
Suppose $X_1$ and $X_2$ are two independent random variables that follow a uniform distribution and $0\leq x_1 \leq k$, $0\leq x_2 \leq k$, where $k$ is a positive constant. Now I want to find the ...
- 25
0
votes
1
answer
72
views
Probability of observing at least one accident in a $30$ period.
You are standing at a bus stop watching cars go by. The probability of observing at least one accident in an hour interval is $3/4$
What is the probability of observing at least one accident within ...
- 1
1
vote
1
answer
111
views
conditional expected value of order statistics
Let $𝑋_{(1)}$,...,$𝑋_{(𝑛)}$ be the order statistics of a set of $𝑛$ independent uniform $(0,1)$ random variables. I need to calculate
$$E[(X_{(2)}-(X_{(1)}+a))^{+} | X_{(1)}<a],$$
where $Y^{+}$ ...
- 11
0
votes
1
answer
95
views
Probability distribution of binomial variable multiplied by a uniform variable [closed]
Given is a binomial variable with success rate $p$, and on success, you receive anywhere between $X_1$ and $X_2$ dollars, uniformly distributed. On failure, you receive nothing.
From what I've ...
- 13
1
vote
1
answer
27
views
Percentage of non picked balls with replacement
Experimenting with numpy and generating random arrays of integers (discrete uniform distribution) I noticed something but can't explain how it would be calculated.
I generate a random sample of ...
- 13
1
vote
0
answers
42
views
Can we tell whether two random variables are independent just by looking at their joint PDF?
Question
Let $X$ and $Y$ be independent uniform random variables on $[1, 2]$. Define $U = \min\{ X, Y \}$ and $V = max \{X, Y \}$.
Show that the joint PDF of $U$ and $V$ is given by
$$ f_{U,V}(u,v) = ...
- 167
2
votes
0
answers
64
views
Distribution function of $X-Y$, where $X,Y$ are uniformly distributed
I want to compute the distribution function of $X-Y$, denoted by $F_{X-Y}$, where $X,Y$ are indepedently uniformly distributed on $[-1,1]$.
\begin{align*}
&F_{X-Y}(z)=\begin{cases}
0,&z\leq -2\...
- 4,298
1
vote
1
answer
52
views
Cumulative distribution function of $2X-3Y$, where $X,Y$ are uniformly distributed
Let be $X,Y$ two random variables that are independently uniformly distributed on $[1,3]$. Compute the pdf and cdf of the random variable $Z:=2X-3Y$. (Hint: use convolution formula)
My approach:
We ...
- 4,298
2
votes
0
answers
31
views
Distribution of concominant order statistics
Motivating problem: We have $n$ students writing a mock test, and a day after, they write a final test. Let $X_i$ represent the grade (continuous from $0$ to $\infty$) of $i$-th student from the first ...
- 722
0
votes
1
answer
100
views
Expected value of $X-Y$, both uniformly distributed
Let be $X,Y$ two independent uniformly distributed random variables on $[0,1]^2$. Calculate the expected value $\mathbb{E}(X-Y)$.
Actually, it seems pretty easy as we can simply use the properties of ...
- 4,298
1
vote
0
answers
36
views
Generate uniform random numbers with a certain dependency structure
Let's assume I wanted to sample from a $\mathcal N(\boldsymbol 0, \boldsymbol\Sigma)$ distribution, where $$\boldsymbol\Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1\end{pmatrix}$$ for some $\...
- 165
0
votes
1
answer
138
views
Expected value of maximum of two D20 rolls
I'm trying to work through a similar version of this problem. The idea is that you have a fair, 20-sided die, and $n$ turns. You are trying to decide how many times to roll it (each roll costs a turn)...
- 35
0
votes
1
answer
74
views
Rosenthal First Look at Rigorous Probability Theory Uniform Construction from Bernoulli
I am reading the mentioned book by Rosenthal, in particular pg 74 where a uniform is constructed from an infinite sum of i.i.d. Bernoulli, i.e., $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ with
$$U=\sum_{k=1}^{\...
- 1,286
0
votes
1
answer
37
views
Find the probability that $P( \frac{1}{\sqrt{2}}\leq (\frac{X^2}{Y^2})\leq\sqrt{2})$ [duplicate]
PROBLEM: Given that $X$, $Y$ are indepedent uniformly distributed r.vs's over range $[0,1]$ with their joint pdf given by: $f_{X,Y}(x,y)=1$
find $P( \frac{1}{\sqrt{2}}\leq (\frac{X^2}{Y^2})\leq\sqrt{...
- 544
0
votes
1
answer
38
views
4
votes
2
answers
153
views
Expected Value of a DnD/Baldur's Gate Feat
The Baldur's gate feat Savage Attacker lets you reroll an attack, and choose the better of the two attacks.
I am attempting to calculate the expected benefit of this feat. Assume, for the sake of ...
- 1,456
3
votes
0
answers
162
views
I need a rigorous explanation of Shoemake method to sample uniformly from the group of unit quaternions
I know and understand the subgroup algorithm to sample from uniform distribution on the
rotation group $SO\left( 3 \right)$, following the following steps:
sample $\theta_{1}$ from $\text{Unif}\left[ ...
- 620
0
votes
0
answers
22
views
Finding the cumulative distribution function of a maximum function, based on a uniform distribution [duplicate]
Suppose I have a uniform distribution
$X$ ~ U($0, θ$).
I have an estimator which pertains to the random variable of the largest value among my random sample. We can express this as
max$(X_1, X_2...X_n)...
- 33