0
votes
1answer
20 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...
0
votes
0answers
24 views

Relationship between quotient of sum of exponentials and uniform distributions

Let $X$, $Y$ and $Z$ be iid with $P(X>t)=e^{-t}$ for $t>0$. Let $U$, $V$ be independent uniform on $[0,1]$. Let $A=\min(U,V)$ and $B=\max(U,V)$. Show that $(A,B),$ and $(X/(X+Y+Z), ...
0
votes
1answer
73 views

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ )

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ ). Find.... a) What is the distribution of $Y$, given $X = x$? b) What is $f(x,y)$? Distribution of $(X,Y)$? c) $f_Y ...
0
votes
2answers
67 views

what is the conditional probability $P(X+Y=2|X-Y=0)$?

Consider two independent random variables $X$ and $Y$ with identical distributions The variables takes values $0,1, 2$ with probabilities $\frac12,\frac14,\frac14$. what is the conditional ...
3
votes
1answer
40 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
0
votes
1answer
41 views

Expected value of the floor function of a sum of two variables

In a recently published paper I have encountered the following equality. Let $U$ be a random variable uniformly distributed in $[0,1]$ and let $Z$ be a Gaussian variable with mean zero and standard ...
1
vote
1answer
60 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
0
votes
0answers
51 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
0
votes
1answer
39 views

Probability question on uniform distribution

I need help with the following question: A computer in adding numbers rounds each number to its nearest integer. Suppose that all rounding errors are independent and uniformly distributed over ...
1
vote
1answer
33 views

Probability of a certain result obtaioned by throwing an octahedron

Assume having a fair octahedron. We throw it $93$ times and get the following results: $\{33;7;8;1;2;0;5;37\}$ The numbers represent how many times the die fell on side $1, 2,...., 8$. What is the ...
4
votes
2answers
92 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
0
votes
1answer
21 views

Uniform Spinner is spun twice..

A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1]. I'm trying to answer the question what is the joint pdf for V and W. I know that I have to ...
1
vote
1answer
33 views

Can I uniformly sample from $N$ distinct elements, where $N$ is unknown but finite?

I have access to a list of $N$ elements, but the value of $N$ is unknown. The elements arrive one by one, and never repeat. I want to sample $n$ of these elements as uniformly as possible, as I have ...
1
vote
1answer
37 views

What is the probability $P\{X_1 \rm{~is ~largest}\}$? [closed]

Let $X_1,X_2,X_3$ be three independent and mutually identically distributed random variabe with uniform distribution on [0,1]. What is the probability $P\{X_1 \rm{~is ~largest}\}$?
2
votes
0answers
41 views

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$.

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I ask this ...
1
vote
2answers
48 views

Approximate distribution for the sample mean?

A random variable $X$ is said to follow a discrete uniform distribution if its probability function is given by $$p_X(x) = \left\{ \begin{array}{ll}\frac{1}{\theta}, & x = 1, 2, \ldots, ...
0
votes
2answers
74 views

What is the pdf of $X,Y$?

We know that the common pdf of $X,Y$ is constant function, on the triangle $(0,0),(0,1),(2,0)$ (and out of this range the value of the function is zero). What is $f_X(x)$ and $f_Y(y)$? My solution: ...
3
votes
1answer
169 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
0
votes
0answers
10 views

Uniform distribution over an hyper-ellipsoid

Let $\mathbf{X} \in \bf{R}^p$ be a random vector whose elements are uniformly distributed over the hyper-ellipsoid $x^TAx<1$, (where $A$ is a positive-definite matrix). Is it possible to compute ...
4
votes
3answers
50 views

Joint density problem. Two uniform distributions

This is the problem: An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. ...
1
vote
1answer
73 views

Convolution of two Uniformly distributed r.v. ove

Assume a continuous random variable $X$ that is uniformly distributed $\underline{\text{on}}$ a $k$-sphere. For simplicity, lets assume a simple circle with radius $R$ in 2 dimension. Therefore ...
1
vote
3answers
62 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
1
vote
1answer
35 views

Proving that a moment generating function converges pointwise

I have found a moment generating function $M_n$ given by $\cfrac{(1-e^t)e^{\frac tn}}{n(1-e^{\frac tn})}$ if $t\ne 0$ and 1 if $t =0$ How do I prove that $M_n$ converges point-wise to the moment ...
2
votes
1answer
42 views

P.d.f. of $XY$, where $X, Y$ are independent uniformly distributed over $[0,1]$ [duplicate]

I tried to change the variables: Let $U=XY$ and $V=Y$; so then the Jacobian is $1/v$. So joint pdf $g(u,v) = f(x,y)\cdot (1/v) = 1/v$ Would you then integrate over $v$ from $0$ to $1$ to get the ...
1
vote
1answer
18 views

MLE for lower bound of Uniform Distribution

Let $X_1$, $X_2$, . . . , $X_n$ be a random sample from a $Uniform(θ, 1)$ population, where $θ < 1$. (a) Find the MLE $\widehat{\theta}$ of $θ$. (b) Find constants c and d (possibly depending on ...
1
vote
0answers
14 views

Fast uniformity test within a ball.

Assume I have a dataset lies within a ball centered around the origin, I want to test the uniformity of the point distributed in the ball. In addition, I have all the distances to the origin computed ...
1
vote
2answers
33 views

Standard Uniform Distribution

I am trying to show that a random variable $X_2$ has a standard uniform distribution. I have: $\alpha \subset(0,1), X_1 \sim U[0,1],$ and $X_2 = \begin{cases} X_1 &\mbox{if } X_1< \alpha ...
2
votes
1answer
47 views

Finding correlation of Max and Min of two IID random variable in U[0,1]

I have a hw problem and can't figure out how to do it. Basically, $X,Y$ are iid $U[0,1]$, we need to find the correlation between max$(X,Y)$ and min$(X,Y)$. My thought is to find the pdf of ...
0
votes
0answers
41 views

Expectation and Variance of a Discrete Uniform Distribution using the Probability Generating Function and Cumulant Generating Function

Hi I just derived the MGF of a discrete uniform distribution and found it to be: [e^t - e^t(m+1)]/(1 - e^t)m and the pgf is ...
2
votes
0answers
38 views

Expected value of winning with a Random Uniform Variable

I have this small homework I don't quite understand. The problem is formulated as a game. (Who wants to be millionaire) You start with 1£. money = 1.0£ You can choose to quit at anytime So you can ...
1
vote
1answer
62 views

Expected value of multiple random variables, uniform distribution

Suppose that the random variables $X_1,\dotsc,X_n$ form a random sample of size $n$ from the uniform distribution on the interval $\left[0, 1\right]$. Let $Y_1 = \min\left\{X_1,\dotsc,X_n\right\}$, ...
8
votes
3answers
209 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
1
vote
2answers
77 views
-2
votes
1answer
40 views

A basic question on uniform distribution [closed]

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
36 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
42 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
28 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
44 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
47 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
40 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
56 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
33 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
77 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
46 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
31 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
87 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
46 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
49 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
108 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
70 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 ...