For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.
Questions tagged [uniform-distribution]
2209 questions
78
votes
5 answers
Density of sum of two independent uniform random variables on $[0,1]$
I am trying to understand an example from my textbook.
Let's say $Z = X + Y$, where $X$ and $Y$ are independent uniform random variables with range $[0,1]$. Then the PDF
is
$$f(z) = \begin{cases}
z & \text{for $0 < z < 1$} \\
2-z & \text{for $1 \le…
Zhulu
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61
votes
3 answers
product distribution of two uniform distribution, what about 3 or more
Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.
What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?
What if there are 3; $Z_3 = X_1…
lulu
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57
votes
3 answers
Probability that a quadratic equation has real roots
Problem
The premise is almost the same as in this question. I'll restate for convenience.
Let $A$, $B$, $C$ be independent random variables uniformly distributed between $(-1,+1)$. What is the probability that the polynomial $Ax^2+Bx+C$ has real…
Hungry Blue Dev
- 1,659
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- 31
54
votes
1 answer
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq…
Jack D'Aurizio
- 338,356
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- 353
- 787
30
votes
2 answers
Expectation of the min of two independent random variables?
How do you compute the minimum of two independent random variables in the general case ?
In the particular case there would be two uniform variables with a difference support, how should one proceed ?
EDIT: specified that they were independent and…
BlueTrin
- 555
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29
votes
0 answers
The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms
PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms.
It suffices to show that the terms of the sequence
$$\,b_n=\mathrm{e}^n\,\mathrm{mod}\, 2,\,\,\,n\in\mathbb N,$$
are…
Yiorgos S. Smyrlis
- 78,494
- 15
- 113
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22
votes
2 answers
Answered: With what probability do $4$ points placed uniformly randomly in the unit square of $\mathbb{R}^2$ form a convex/concave quadrilateral?
I have this problem that I've struggled with for a while. If you place $4$ points randomly into a unit square (uniform distribution in both $x$ and $y$), with what probability will this shape be convex if the $4$ points are connected in some order?…
Leonhard Euler
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20
votes
4 answers
Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$
I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, $\mathcal{L}(a,b)=\frac{1}{(b-a)^n}$ is constant, how do I find a maximum? Would…
Spine Feast
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- 62
19
votes
2 answers
joint distribution, discrete and continuous random variables
This may be trivial, but if X is a random variable uniformly distributed over $[0,1]$ and Y is a discrete random variable such that $\mathbb{P} (Y=y_1) = \lambda \in (0,1]$ and $\mathbb{P} (Y=y_2) = 1 - \lambda$. Now I am seeking to compute the…
Mia
- 491
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- 3
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17
votes
2 answers
Picking points on a sphere at random
Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that all the points lie within any hemisphere on the…
anurag anshu
- 367
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- 7
16
votes
2 answers
Calculate $\lim_{n \to\infty}\sqrt[n]{\{\sqrt{2}\}\{2\sqrt{2}\}\{3\sqrt{2}\}\cdots\ \{n\sqrt{2}\} }$
$$\text{Calculate :}\lim_{n \to\infty}\sqrt[n]{\{\sqrt{2}\}\{2\sqrt{2}\}\{3\sqrt{2}\}\cdots\{n\sqrt{2}\} } . $$
Note:
Weyl's equidistributed criterion. The following are equivalent:
$$x_n\quad\text{is equivalent modulo 1}$$
$$\forall~…
Rick Sanchez C-666
- 759
- 3
- 14
15
votes
2 answers
Complete Statistic: Uniform distribution
Take a random sample $X_1, X_2,\ldots X_n$ from the distribution
$f(x;\theta)=1/\theta$ for $0\le x\le
\theta$.
I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete.
Now, I know I should multiply the sample distribution of $Y$ and multiply it…
Bree
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15
votes
0 answers
Calculate $\pi$ from digits of $\pi$
With a random normal distribution $\pi$ can be calculated with help of the PDF (probability density function). The method below apparently shows $\pi$ can be determined with random digits $[0,1,2,3,4,5,6,7,8,9]$. If this is correct and $\pi$ is a…
OOOVincentOOO
- 530
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- 19
15
votes
2 answers
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!
Perdue
- 311
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14
votes
3 answers
Probability the three points on a circle will be on the same semi-circle
Three points are chosen at random on a circle. What is the probability that they are on the same semi circle?
If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ and $c_2$. then it will imply that…
kangkan
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