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
35 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
-2
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0answers
17 views

Conditional Distribution of Ordered Statistics [on hold]

Let $X_1,...,X_n$ be the order statistics of a set of $n$ independent uniform (0,1) random variables. Find the conditional distribution of $X_n$ given that $X_1=s_1,...,X_{n-1}=s_{n-1}$. I honestly ...
0
votes
2answers
39 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
1
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0answers
12 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
-1
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1answer
30 views

Demonstrate uniform continuous distribution using tangible items?

What is the best way to explain "equally likely" in continuous uniform distribution to an audience using tangible or everyday items?
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2answers
37 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
2
votes
1answer
30 views

Stoppage time for sequence of uniform random numbers with a recursively shrinking domain

Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which ...
0
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1answer
23 views

How were the bounds of integration changed here in this probability problem on uniform distributions?

I have the following problem and solution: I don't understand how the bounds of integration were changed from 0 to 1, to $x^2$ to 1. I see where $1/\sqrt{y}$ was substituted in for $f(x|y)$ and 1 ...
1
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1answer
20 views

calculating probabilities about the sum of dependent discrete uniforms

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations ...
0
votes
2answers
38 views

Probability Question About Uniform Random Variables and Median

Let U, V, W ∼ Uniform(0, 1) be independent. Find the probability that the median (i.e., the second smallest) of these three random variables lies in the interval (1/4, 3/4). I cannot figure out what ...
0
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1answer
138 views

uniformly distributed random variables

Ram and Shyam wanted to meet at a park about 12.30 P.M.. If Ram arrives at a time uniformly distributed between 12.15 P.M. to 12.45 P.M. and if Shyam independently arrives at a time uniformly ...
1
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2answers
42 views

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 ...
2
votes
2answers
50 views

Mean of the difference between uniform random variables.

I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then ...
0
votes
1answer
38 views

interval learning unbiased estimator

Suppose that I draw $n$ points uniformly at random from $\mathcal{U}(-a,a)$ for some $a\in\mathbb{R}_+$. Denote this set of points $\{X_i\}_{i=1}^n$. Now for some $-a < x_1 < x_2 < a$, let a ...
2
votes
1answer
35 views

Indicator function and expectation

Consider a random variable $X$ uniformly distributed over $[0,1]$ and the indicator function $\mathbb{1}_{X \geq \tilde{x}} $ equal to one if $X \geq \tilde{x}$ and zero otherwise. We know that ...
1
vote
1answer
21 views

Conditional Probability Distribution for two Discrete Uniform Random Variables with given Correlation Coefficient

I consider a problem with two random variables $X, Y \sim Unif\{a,b\}$, for which I want to set a correlation coefficient $Corr(X,Y)=\rho$. Now, I am interested in the conditional probability mass ...
1
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2answers
39 views

How to prove that equation over probabilities has unique solultion or find counterexample?

Given equations: $$ \prod_{i=1}^n p_i = \prod_{i=1}^n (1-p_i)= \frac{1}{2^n} $$ where $p_i\in (0,1), i=\overline{1,n}$. Is it true that this system has unique solution $p_1=p_2=\ldots=p_n=\frac12$ ...
0
votes
4answers
57 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that ...
2
votes
1answer
16 views

Calculating power of a Hypothesis Testing Problem based on Uniform distribution

Consider the problem of testing $H_0:a=0$ against $H_1:a=1/2$ based on a single observation X from U(a,a+1). The power of the test "Reject $H_0$ if $X>2/3$" is (A)1/6 (B)5/6 (C)1/3 (D)2/3 ...
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0answers
25 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
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0answers
30 views

Continuous Uniform Distribution

Suppose $X$ follows a continuous uniform distribution from 1 to 5. Determine the conditional probability $P(X > 2.5 | X \le 4)$ I am not sure I know how to do ...
0
votes
1answer
17 views

Order statistics and biased estimators

Would someone be able to check my work on this question: Suppose that $Y_1,Y_2,...,Y_5$ is a random sample from a uniform distribution over the interval (0, theta). Determine if the followng ...
3
votes
1answer
82 views

Expectation of maximum of two independent random variable with known distribution [closed]

Assume $X$ and $Y$ are two random variables such that $X\sim \textrm{Unif}(0,1)$ and $Y=e^{-t}\times a $ where $t\sim \mathrm{Exp}(\lambda)$ and $a\sim \textrm{Unif}(0,1)$. What is ...
4
votes
2answers
77 views

Sum of discrete and continuos random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are ...
2
votes
1answer
60 views

Two people meeting, expected time of waiting

$A$ and $B$ are supposed to meet. $A$ arrives in a randomly chosen (uniform distribution) moment between $2$ and $3$ pm. $ B$ arrives at $2$ pm with probability equal to $0,5$ and in a randomly ...
0
votes
0answers
26 views

Separating a random variable

Let α and β be random variables uniformly distributed from 0 to 1. Let λ= k1α -c1α + k2β -c2β. Let x be the random variable that is uniformly between ...
0
votes
1answer
23 views

Expected length of shortest interval containing numbers drawn at random

A random idea: If you draw $n$ numbers uniformly at random from $[0,1]$, what is the expected length $L_n$ of the shortest interval that contains all but one of them? Clearly, we have $$L_2 ...
0
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0answers
47 views

Relationships between Uniform and Pareto Distributions

If $X$ is uniformly distributed over $(a,b)$ and $Y$ is pareto distributed with parameters $(min,c)$, what is the distribution of Z in the following cases? (a) $Z = X + Y$ (b) $Z = XY$ (c) $Z = ...
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0answers
53 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...
2
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0answers
56 views

Question on uniform distribution of points on a sphere.

Let N points be uniformly distributed on the surface of a unit sphere $S^2$. What is the probability that every spherical cap of area A contains at least one point? The area $A$ depending on the ...
0
votes
1answer
28 views

Uniform Distribution $Y:=(X+1)^2$

$X \sim uniform[0,4]$ Another probability variable $Y$ is defined as $Y:=(X+1)^2$. I'm searching for the CDF of $Y$. Thing's I already know: If $W:=X+1$ then $W \sim uniform[1,5]$ $Y=W^2$, so ...
0
votes
1answer
48 views

distribution of cosine of uniformly random variables

let us consider following data and here is its chart generally cosine is not linear function,therefore cosine of uniform variables should not be uniform as well,because if $x=cos(y)$ ...
0
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0answers
14 views

A line segment with continuous distribution

$ C $ is a point on a $ AB $ straight line of length $ k $. If the distance $ AC $ is a random variable $ x $ with continuous uniform distribution, evaluate the probability that the difference in ...
2
votes
1answer
35 views

Expected revenue in first-price auction with budget constraint drawn uniformly between [0,1]

I am trying to understand an example from the article "Standard Auctions with Financially Constrained Bidders" Che & Gale (1998) - Review of Economic Studies. Two buyers each value an object at ...
2
votes
1answer
63 views

Probability of average distance from origin of unit circle less than half

Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than half?
2
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1answer
260 views

Show that there is no discrete uniform distribution on N.

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?
2
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2answers
52 views

The Hurried Duelers brainteaser

This question is similar as this other one asked in the forum, but I am trying to give it a different twist. Unfortunately, I am not getting to the same answer, so there might be something wrong in my ...
1
vote
1answer
25 views

Determine the Hypergeometric probability function using sample space in which the selection is ordered

I'm unable to think through this question please help. Suppose our sample space distinguishes points with different orders of selection. For example suppose that $S =\{SSSSFFF\ldots\}$ consists of ...
2
votes
2answers
76 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
votes
1answer
21 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
3
votes
1answer
60 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
5
votes
1answer
43 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
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0answers
28 views

sum of random variables?

$x\to$ uniformly distributed on $(0,1)$ $y\to$ uniformly distributed on $(0,2)$ $z\to$ uniformly distributed on $(0,4)$ What is the probability that $2x+3y < z$? I tried to do it geometrically ...
1
vote
1answer
38 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
1
vote
1answer
26 views

expectation of uniformly distributed $n$ number of samples

I am trying to fine the expectation: $E((x_1+ x_2+ \cdots +x_n )^2)$ as a function of $n$ where all $x_1$ to $x_n$ have uniform distribution $U(0,1)$. I can do if there is only $x_1$ and $x_2$ but ...
0
votes
1answer
13 views

Comparing Uniform Random Variable

X,Y and Z are uniformly distributed random variable on (0,1) What is the probability that X+Y>Z? I tried to do it geometrically and find the volume x+y in the given limits ie 0 to 1 for both x and y. ...
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votes
2answers
113 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...
0
votes
2answers
55 views

How to find $\lim_{n\to \infty} P(a≤(X_1X_2…X_n)^{-n/2}e^{n/2}≤b)$ where $X_1,X_2,…,X_n \sim U[0,1]$?

I am trying to calculate $$\lim_{n\to \infty} P(a≤(X_1X_2...X_n)^{-n/2}e^{n/2}≤b)$$ in terms of $a,b$, where $$X_1,X_2,...,X_n \sim U[0,1]\,\,\,\,\,\,\,(i.i.d.)$$ and $$0≤a<b$$ My attempt is to ...
-1
votes
2answers
47 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
0
votes
1answer
25 views

If the side length of a square follows uniform distribution, how to find the mean and variance of its area?

A square has side of length $X$ cm, where $X\sim U[4,10]$. Find the mean and variance of the area of the square. I understand how to get the mean and variance for the length of each side, but simply ...