-1
votes
1answer
28 views

A basic question on uniform distribution [on hold]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
4
votes
1answer
27 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
0
votes
2answers
32 views

Explanation for the uniformity of the distance between a Gaussian variable to its nearest integer?

earlier I asked the question Expected distance for a gaussian variable to its nearest integer. and got a good answer. The expected distance is highly close to $1/4$, which is very similar to the ...
1
vote
1answer
23 views

Distribution of a uniform random variable with random endpoint

Let $Y \sim U[0,k]$, where $0 < k < \infty$ and $U$ is a continuous uniform distribution. Now let $X \sim U[0, Y]$. What is the distribution of $X$? Is it possible to express in terms of some ...
1
vote
1answer
39 views

Variance of a polynomial series from a uniform distribution

I intend to derive the variance of $Z$: $$Z \equiv \alpha_0+\alpha_1X+\alpha_2X^2+\dots + \alpha_MX^M = \sum_{m=0}^{M}\alpha_mX^m $$ for some $0 < M < \infty$ where each $\alpha_m \in ...
0
votes
0answers
39 views

sum of two independent uniform random variables question

Let ܶ$T_1$ and ܶ$T_2$ be random times for a company to complete two consecutive steps in a certain process. $T_1$ and ܶ$T_2$ are measured in days and their joint probability density function is ...
2
votes
2answers
33 views

Difference Uniform rv's

Let $U_{1}\sim U(0,1)$ be a standard uniform random variable. Is $U_{1}-U_{1}$ uniformly distributed? I've been trying to work this out as follows: Let $A,B$ be rv's $$P(A-B\leq ...
0
votes
0answers
42 views

Uniform Distribution Probability

A manager of an apartment store reports that the time a customer on the second floor must wait after calling the elevator has uniform distribution ranging between 0 and 6 minutes. Assuming that it ...
3
votes
2answers
26 views

Independent two random variables with uniform density

Let $X \sim U[-1,1]$ and $Y=X^2$. Show that $X$ and $Y$ aren't independent. Of course we have $F_X(x) = \frac{x+1}{2}$ and $F_Y(y) = \frac{\sqrt{y}+1}{2}$. But how can I find $$F_{XY}(x,y) = \Pr(X ...
0
votes
1answer
69 views

Expected value with two random variables

A line segment AB of length 1m is broken in two at a random point P where the length of AP has the following probability density function: $f(x)=6x(1-x), 0<x<1$ A point Q is uniformly selected ...
2
votes
3answers
36 views

How do I calculate this expected value?

The problem is as follows: Six players draw, one after another and independently, a number uniformly distributed on $[0,1]$. A player is called a recordist if he draws a number that is larger than ...
1
vote
1answer
22 views

If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability X value is greater than Y value

Suppose $X$ and $Y$ are continuous uniform random variables. If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability that a random $X$ value is greater than a random $Y$ value.
0
votes
0answers
64 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
0
votes
0answers
32 views

What is the conditional distribution of this random vector?

Let us have random vectors $X_1, \dots, X_N$ which are identically independently uniformly distributed in the $n$-dimensional unit hyperbox $[0; 1]^n$. Let $c = (0.5, \dots, 0.5)$ be the center of ...
0
votes
2answers
38 views

Distribute a population size based on fractions using random number generator drand48()

I have a population size of say 5000 people. Every person belongs to either A, B, C or D category. I want to split the population as per a given fraction provided by user. for example, 99% of A, 0.4% ...
2
votes
2answers
70 views

$X$ and $Y$ are uniformly ditributed on $(0,1)$. distribution of $\max(X,Y)/\min(X,Y)$

Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution ...
0
votes
2answers
66 views

Calculate expectation and variance

Let $(X_n)$ be a sequence of independent RVs which are uniformly distributed on $[0,1]$ interval. For $0<x\le 1$ we define $$N(x):=\inf\{n:X_1+\dots+X_n\ge x\}.$$ Show that $$\mathbb{P}(N(x)\ge ...
0
votes
0answers
63 views

Expected Value - Uniform distribution over infinite interval

Question: The probability that an error is introduced into a packet is $\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of ...
0
votes
2answers
50 views

Probability with multiple uniform distributions

Question: Two sources output a number at equal rates. The output from source A is uniformly distributed between 100 and 199, and the output from source B is uniformly distributed between 50 and 249. ...
2
votes
2answers
52 views

Calc expected value of 5 random number with uniform distribution

Assume we have a random numbers $\sim U(0,100)$. Then the expected value of that number will be: $\int_{0}^{100} \frac{x}{100}$ = 50.5 Now assume we have 5 random numbers $\sim U(0,100)$. How can I ...
1
vote
3answers
43 views

uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as Mx(t) = ((e^(tb)-e^(ta))/t(b-a) However I know Mx(0)=1 but I can't get my head around how this is possible as if t=0, ...
2
votes
1answer
85 views

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 $c_1$ ...
1
vote
0answers
44 views

Prove that $E(X_1 \dots X_n)^\frac{1}{n} \leq (EX_1 \dots EX_n)^\frac{1}{n}$, with $X_i$ uniform distribution

Let $X_1, \dots , X_n$ i.i.d. uniformly distributed random variables with $f(x) = 1_{(0,1)}(x)$, $x \in \mathbb{R}$. Let $\Pi_n = (X_1 \dots X_n)^\frac{1}{n}$ and $M_n = \max \{ X_1, \dots , X_n ...
2
votes
1answer
40 views

Finding the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$

I'm trying to find the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$ Here $U$ is the uniform distribution. The method I use i to introduce an auxilary variable $W=X$ and then use ...
4
votes
1answer
99 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
1
vote
1answer
30 views

Simple probability problem.

$X$ ~ $U(0,1)$ and $Y$~U$(0,1)$ are two indenpendent variables. Get Pr ( Y > X). NOW what i don't understant in this problem is how you set the limits of integration. I heard that you must set ...
0
votes
1answer
96 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
1
vote
1answer
62 views

Finding density function for uniform distribution

Can anyone help me set this up correctly, please: John is going to eat at at McDonald's. The time of his arrival is uniformly distributed between 6PM and 7PM and it takes him 15 minutes to eat. Mary ...
2
votes
1answer
176 views

Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, ...
0
votes
1answer
196 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
0
votes
1answer
95 views

Density of uniformly chosen random point inside triangle

Imagine the triangle inside of the points $(0,0), (0,1)$ and $(1,0).$ Let $(X,Y)$ be a uniformly chosen random point from the triangle. Then find the joint density of $X$ and $Y$. The answer is ...
0
votes
1answer
54 views

Bivariate and Multivariate Probability Distributions

For my homework for Bivariate and Multivariate Probability Distributions section, I encounter the terms joint density, joint distributed random variable, joint probability, uniform distribution, when ...
0
votes
2answers
133 views

Uniform distribution over the unit circle

Suppose that $U$ and $V$ are two independent uniform $(-1,1)$ random variables. Any hints on how I can show that their conditional distribution, given $U^2 +V^2<1$ is given by the uniform ...
1
vote
1answer
71 views

How to find the pdf of the minimum of absolute differences of Uniform distributions.

Let $X_1$,$X_2$ and $X_3$ are independent random variables that are uniformly distributed over $(0;b), b>0$. What is the probability density function of z=min($Y_1$,$Y_2)$, where $Y_1=|X_1-X_2|$ ...
2
votes
1answer
31 views

Uniformly Distributed ingredients

Suppose we need to make a dish that has three ingredients A, B and C. All are distributed uniformly between [0, 2], [0, 2], [0, 1] respectively. To create the dish, we need 1/4 of A, 1/4 of B and 1/8 ...
0
votes
1answer
28 views

Finding the joint density of two random variables

Suppose (X,Y) is uniformly distributed over the region { (x, y) : 0 < x < y < 1 }. Find the joint density of (X, Y). I started out by drawing the unit square and filling in the area where 0 ...
0
votes
1answer
44 views

Finding probability of uniform random variable given a condition with another random variable

Suppose X and Y are independent and uniformly distributed on the unit interval (0,1). Find: $$P[Y>\frac{1}{2}\,|\,Y>1-2X]$$ How I approached it was to find the area where $Y > 1 - 2X$, and ...
0
votes
1answer
123 views

Probability of maximum of 2 uniform random variables

The random variables X and Y are independent, each with the uniform distribution on [−1, 1]. Find: $$P[max (X,Y) >0.5]$$ Apparently there is an easy approach without integration, but I am having ...
1
vote
0answers
362 views

What is the expected time you have to wait until the first bus comes?

 three buses, bus A, B, and C come to a bus stop every hour. The time at which each bus arrives at the stop is distributed as a uniform random variable, i.e., TA,TB,TC ∼ Unif[0,1] hours. The ...
3
votes
1answer
58 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
1
vote
1answer
51 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
3
votes
4answers
261 views

Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$

What is the probability that $a > b + c$? $a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
2
votes
0answers
502 views

Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
1
vote
0answers
67 views

Uniform distribution

You arrive at a bus stop at 10'0 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. what is the probability that you wait longer than 10 minutes? if at ...
0
votes
1answer
106 views

Linear Combination of Min and Max of Uniform Random Variables

Let $p$ and $q$ be uniformly distributed on $[0,1]$. Define $x=\min\{p,q\}$, $y=1-\max\{p,q\}$ and $z=1-x-y$. What are the distribution functions of $x$,$y$ and $z$? I've got $F_X(x) = 1 - (1-x)^2$ ...
0
votes
1answer
30 views

Finding variance .

Suppose that $f : [0, 1] → [0, 1]$ and we wish to estimate $$I = \int_{0}^{1} f(x) dx$$ Using the hit-and-miss method, we obtain the estimate $$\hat I_{HM}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ where ...
0
votes
1answer
117 views

Plot the cdf and simulate a random variable (rv) with this cdf using the inversion method.

Consider the continuous random variable with pdf given by: $$f(x) = 2(x − 1)^2;\quad 1 < x ≤ 2$$ $$f(x) = 0;\quad \text{otherwise}$$ Plot the cdf for this random variable. Show how to simulate ...
0
votes
2answers
282 views

Uniform Distribution : pdf & inverse cdf

$X\sim U(1,3)$. Verify that X has cdf $F_X(x) = 2(x − 1)$ for $x \epsilon(1, 3)$ and thus that $F^{−1}_X (y) = 2y +1$ for $y \epsilon (0, 1)$. My attempt for ...
0
votes
1answer
171 views

Question regarding finding Joint distribution of two random variables

I have a question regarding finding the following joint distribution. Let $p \sim U[0,1]$, standard uniform distribution. The random variable $X$ is defined as $X = 2$ with probability $p$ and $X ...
1
vote
1answer
883 views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...