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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1answer
26 views

Properties of the solution of a linear system with random equations

$x_i$ is drawn from $\mathrm{unif}(a,b)$, $y_i$ is drawn from $\mathrm{unif}(c,d)$. $x_i$ are independent from each other. $y_i$ are independent from each other. $x_i$ are independent of $y_i$. $i$ ...
1
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0answers
22 views

Repetitions required to choose from a list

Say I have a list of $n$ objects and I randomly choose one item from the list, but did not remove it. What is the method for calculating the probability I have chosen $x$ different items in the list ...
2
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0answers
46 views

Distribution of $1/U^2$ where $U$ is uniformly distributed on $(-1, 1)$

Suppose $U\sim \mathrm{Uniform}(-1, 1)$. Let $Y =1/{U^2}$. What is the distribution of Y? Here is what I have: $$ \begin{aligned} Y \in [1,\infty)\\ P(Y <y) = P\Big(\dfrac{1}{U^2} < y \Big)\\ =...
-3
votes
1answer
45 views

Help in probability, Difficult Question:// [closed]

Upon testing 80 resistors manufactured by a certain company, it is found that 15 resistors failed to meet the tolerance design specifications a) Construct a 92% two-sided confidence interval for ...
-2
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2answers
33 views

Generating a random variable from a uniform random variable [closed]

I have no idea how to go about doing this. Any help would be much appreciated.
0
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0answers
22 views

As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? [duplicate]

both for the situation where elements in the matrix are randomly assigned any number and where elements are assigned a value uniformly distributed between 0 and 1. Thanks so much! (This is not a ...
0
votes
1answer
37 views

Uniform Distribution Question - Help Needed

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
1
vote
1answer
40 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
0
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1answer
18 views

Maximum-Likelihood-estimator for number of marbles

Let there be n marbels in a box. Each has a unique number from 1 to n written on it. You pick a marble, write down the number and put it back. This process is repeated N times. Give a maximum-...
0
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1answer
50 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
-1
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2answers
33 views

Sum of uniform random variable and non-uniform random variable [closed]

Let $G=\mathbf{Z}/p \mathbf{Z}$ where $p$ is prime, $X\in G$ be a uniform random variable and $Y\in G^{*}$ be any random variable. Is it possible to have $Z=X+Y \in G$ with a uniform distribution? ...
1
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1answer
22 views

Radio communications system - Uniform density question [closed]

Full question: In a radio communications system, the phase difference $X $ between the transmitter and receiver is modeled as having a uniform density in $[—\pi, +\pi]$. Find $P(X \le 0)$ and $P(X \le ...
1
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1answer
74 views

Random Variables in a Uniform Probability Space

Suppose that $\Omega = \{1,2,3,4,5,6\}$ is a uniform probability space. Now, let $X(\omega)$ and $Y(\omega)$, for $\omega \in \Omega$, be random variables defined as: $$\begin{array}{|c|c:6c|} \...
0
votes
2answers
25 views

Simulation methods and generating random variables

Twenty aircraft are sent to bomb a target that is rectangular in shape. It has dimensions 150m by 50m. Each aircraft makes a bombing run along the horizontal x axis and drops one bomb. The point ...
1
vote
1answer
46 views

What is E[X]? What is Var(X)?

The number of accidents X that a person has in a given year is a Poisson random variable with mean Y . However, Y ∼ Uniform ([2, 4]). Calculate: (a) E[X] (b) Var(X) Extra My understanding of the ...
1
vote
1answer
77 views

Probability that at least two of three uniform random variables~[0 1.5] add up to >2 [closed]

There's a problem I've been stuck on for a while regarding the sum of two uniformly distributed, independent random variables. The problem goes like this: You find some old batteries in a drawer. ...
0
votes
1answer
49 views

How could I show that some random variables have same distribution?

Let X1, . . . , Xn, be random variables uniformly distributed over the interval [a, b]. Let Y1 < Y2 < · · · < Yn be the same values in sorted order. Let Y0 = a and Yn+1 = b. Show that ...
0
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0answers
14 views

Analytical expression for expected the n-th and (n-1)th order statistic of minimum of two uniform random variables

I have two independent random variables $X_1 \sim Uniform(b,1)$ and $X_2 \sim Uniform(0,1)$, where $0<=b<1$. Their CDF's are: $$ F_{X_1}(z) = \begin{cases} 0 & z<b \\ \frac{z-b}{1-b} &...
0
votes
1answer
65 views

Assuming the two people wait for each other, what is the expected waiting time?

Two people agree to meet at a restaurant. Assume their arrival times are independent and uniformly distributed on the one hour interval from 1:00–2:00 p.m. Assuming the two people wait for each other, ...
0
votes
1answer
25 views

Box-Muller method to Polar Marsaglia scheme

I have just learned the Box-Muller method for generating normal random values. My notes then consider the Polar Marsaglia method, which is more efficient than Box-Muller. In Box-Muller: $$X=\sqrt{-2\...
0
votes
1answer
21 views

How do we know the pdf of $X_{max}$ for a uniform distribution? [closed]

How do we know the pdf of $X_{max}$? We have a sample of size $n=6$ from a uniform distribution of the interval $[0,\theta]$. Let $\hat{\theta}=X_{max}$ be the estimator of $\theta$. (a) Given $\...
0
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0answers
33 views

Tukey's symmetrical lambda distribution

U ~ Uniform(0,1) $$Z_\lambda = \frac{U^\lambda-{(1-U)}^\lambda}{\lambda} $$ I have to find the first four moments and two ($\lambda_1,\lambda_2$) such that they have the same four moments.
1
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0answers
12 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
0
votes
1answer
30 views

What is the conditional distribution function of P(T < z| U=x) where U, V are uniformly distributed on [0,1] and T= Max(U,V)

Let U, V be independent random variables, both uniformly distributed on [0,1].T= max (U, V). 1) What is the joint distribution function of T and U, P(T <= Z, U <= x)? I did the following: P(T ...
0
votes
1answer
34 views

Understanding Bayes' theorem with uniform distribution

Station $X$ begin to transmit a message in $[0,20]$ with uniform distribution, and $Y$ also want to transmit a message in $[6,14]$ with uniform distribution. Assume that transmission takes $2$ ...
1
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1answer
23 views

Integrating PDF of Continuous Uniform RV to get CDF

For a continous uniform RV $$\text{PDF} = \frac{1}{b-a} \text{ for } x\in[a,b)$$ (sorry cant figure out how to add whitespace) and $$\text{CDF} = \int_{-\infty}^x \text{PDF} \, dx$$ so for $x>0$ ...
0
votes
0answers
26 views

Distribution of a function of Discrete Uniform RVs

Suppose $X$ and $Y$ take the values ${0,1,...,N}$ with probability $\frac{1}{N+1}$. What is the PMF of $|X-Y|$? I started it like so: Let $|X-Y| = Z$ $P(Z<z) = P(-z < X-Y<z)$ In the ...
2
votes
0answers
15 views

Targetting the distribution of distances between points

For a certain problem, I need to make distance dependent statistics, but with the constraint that the number of sampling points, $N$, should be kept as small as possible. To be more specific I need to ...
0
votes
1answer
50 views

Find the joint pdf of X and Y for two uniform distributions

Let X have a uniform distribution on the interval $(0,1)$. Given that X = x, let Y have a uniform distribution on the interval $(0,x+1)$. Find the joint pdf of X and Y. Sketch the region where $f(x,...
0
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0answers
20 views

evaluation of this integral arising in probability theory

Let $U_1,...,U_n$ be independent, uniformly distributed RVs on $[0,n]$. Consider the ordered statistics of these, $$U_{1:n} < ... < U_{n:n}$$ So $U_{1:n} = \min\{ U_1,...,U_n\}$ and so on. ...
3
votes
1answer
39 views

Average waiting times

I have the following exercise, which I would like to solve: Company A run buses between New York and Newark, their bus leaves New York every half an hour starting from 0:00, 24h a day. Company B also ...
0
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0answers
21 views

Uniformly Random Polynomial

Hypothesis: all the polynomials and values are defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number. My question is related to information security. Assume (in a protocol) ...
0
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0answers
9 views

Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
0
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0answers
29 views

Convolution of a Pareto and a Uniform distribuion

I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
3
votes
1answer
35 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...
0
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0answers
24 views

Working out probability distribution from given uniform random variable

I'm not able to understand how they got from probability U is less than e to the power of x to e^-x? shouldn't it be e^x instead because we are given in our notes that probability U that's less than u ...
0
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1answer
18 views

does calculating the arithmetic mean of a serie of standar deviation measures make sense?

I'm trying to test if a hash function mantains a even distribution of values. My plan is to generate a set X of hashes and a integer Y symbolizing "slots" to see how many of the hashes maps to the ...
8
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0answers
75 views

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 ...
1
vote
2answers
51 views

A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability B arrives before A?

As the title indicated, A arrives at office at 8- 10 am everyday, B arrives at 9-11 am everyday. Probability that One day B arrives before A? I am bit confused whether to use Poisson (because arrival ...
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2answers
40 views

If $X_1,X_2,X_3$ are three different uniform random variables, calculate $E(X_1 - 2X_2 + X_3)$. [closed]

Suppose that a random variable $X_1$ is distributed uniform $[0,1]$, $X_2$ is distributed uniform $[0,2]$ and $X_3$ is distributed uniform $[0,3]$. Assume that they are all independent. a) Calculate ...
0
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0answers
27 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 \...
2
votes
2answers
46 views

If the area of a square is uniformly distributed, then find the expectation and variance of the side length.

I'm told that the area of a square (denoted $A$) is uniformly distributed across the interval $[15,20]$, I'm then asked to find the expectation and variance of the side length but I can't work out how ...
0
votes
2answers
73 views

Probability that the inpatient guy arrived first

So the question asks: Tom and Jerry set up a meeting at a restaurant. Each one of them, independently of the other, arrives at some random time between 9:00 pm and 10:00 pm (that is, the arrival ...
3
votes
2answers
42 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let $P\in\{0,1\}^{n\...
2
votes
1answer
35 views

Combine discrete uniform distributions to achieve a discrete uniform distribution of a larger range?

How can I effectively combine multiple discrete uniform distributions of a limited range to achieve a discrete uniform distribution of a large range? I.e. given $unif\{a,b\}$ generate $unif\{a,c\}, c&...
1
vote
1answer
75 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
3
votes
1answer
34 views

Is modular multiplication under a prime modulus uniformly distributed?

Let's say that I have a prime, $p$, and an $m \in Z_p^*$. Then, I draw $a \leftarrow Z_p^*$ uniformly at random. Will $am \mod p$ be distributed uniformly over $Z_p^*$?
0
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0answers
35 views

Rewiting an integral in the case of a uniform distribution

I would like to rewrite this equation: \begin{equation} \widetilde{A}(\varphi^*) = \left[\frac{1}{1-G(\varphi^*)}\int_{\varphi^*}^{\infty} A(\varphi)^{1-\sigma}g(\varphi)d\varphi\right]^{1/(1-\sigma)} ...
0
votes
1answer
28 views

Uniformly Distributed Random Variable over an interval

Let $X$ be a random variable uniformly distributed over a nontrival interval $[c,d]$, and let $Y = aX+b$. For what choice of real constants $a$ and $b$ is $Y$ uniformly distributed over [0,1]? How ...
1
vote
0answers
27 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

The Statement of the Problem: Let $ \{ U_i \}$ be a set (sequence?) of iid random variables such that $U_i \sim \text{Uniform}(0,1)$, and define $$ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\...