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1answer
31 views

Pdf of log of ratio of uniform random variables

Suppose $U$ and $V$ are iid $U(0,1)$. I am trying to find the pdf of $W=log(U/(1-V))$. My approach is to suppose $Z = U$ and find the joint density of $(W,Z)$ and get the marginal of $W$. The pdf of $...
3
votes
1answer
213 views

How does the inverse transform method work in discrete r.v.?

In this question How does the inverse transform method work? it's mentioned the general procedure to generate r.v. U <- runif(1e6) X <- qnorm(U) X How ...
0
votes
1answer
38 views

Finding the uniformly most powerful test for hypothesis

Let $\mathbf{X}=(X_1,...,X_n)^T$ is a simple sample where $X$ belongs to exponential distribution family $\mathcal{P}=\{ f(x;\mu,\sigma \}, -\infty<\mu<\infty, 0<\sigma<\infty.$ Density is ...
0
votes
1answer
102 views

How to generate a conditional random variable in R? [closed]

Suppose there is a sample $X\sim N(0,1)$ x<-rnorm(100). If I want to generate a conditional random variable $Y|X\sim U(0,1)$, how can I get this conditional ...
2
votes
1answer
80 views

Joint cumulative distribution of independent random variables

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...
3
votes
2answers
107 views

X is Uniform $[-\theta,\theta]$ what is the distribution of $Y=\frac{1}{x^{2}}$?

X is Uniform $[-\theta,\theta], \theta>0$ what is the distribution of $Y=\frac{1}{x^{2}}$ So I've been working on some transformation questions; however, most of them have been one to one. I am a ...
2
votes
1answer
49 views

probability that matrix $2\times2$ of Random variables is Invertible

Let $X_1, X_2, X_3, X_4$ to be Variables, and let $A$ be the following matrix: $$ \left[\begin{matrix} X_1 & X_2\\ X_3 & X_4 \end{matrix}\right] $$ assume that $X_1, X_2, X_3, X_4$ are ...
1
vote
2answers
1k views

How to find distribution function of sum of 2 random variables that are uniformly distributed? [duplicate]

I am stuck with this tutorial question in one of my stats module and I would greatly appreciate some help: Let $X1$ and $X2$ be independent random variables with $a = 0$ and $b = 1$ i.e. $X1$ and $X2$...
2
votes
2answers
32 views

Getting variance of function of two uniform RVs [duplicate]

Have two independent RV's $X$ and $Y$ sampled uniformly from $[0,1]$ and $C = (X-Y)^2$. Want $V(C$). Rewrote as $V((X-Y)^2) = V(X^2) - 4V(X)V(Y) + V(Y^2)$ but that's too messy. Is it correct to write ...
28
votes
3answers
3k views

Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a ...
3
votes
1answer
67 views

Convergence of sum of exponentially weighted random variables

I don't know if the title is accurate, but I have this problem: I have iid RVs $Y_k$ that has a value from {0,1,...,9} with equal probability. I need to show that $$ X_n = \sum_{k=1}^{n}Y_k10^{-k} $$ ...
4
votes
0answers
1k views

Generate Beta distribution from Uniform random variables

I need to generate random numbers from Beta distribution using random variables from Uniform distribution. If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$, and If $Y_1+Y_2&...
0
votes
1answer
69 views

Independent random variables and survival function

I just started taking survival analysis class and I'm stumped on this question. Let $T_{1},T_{2},...T_{n}$ independent continuous non-negative random variables with survival function $S(t)$ Show ...
3
votes
2answers
76 views

What proportion of the space is taken up by independent discrete uniform variables

If you take $N$ independent uniform random selections from a discrete space with $M$ possibilities (with replacements), then what proportion of the possibilities will have been selected? Formally, ...
3
votes
1answer
66 views

Showing Independence for X and frac(X + Y)

Suppose that we have independent samples $X_1, X_2 \sim \text{unif}[0, 1)^d$. I'm asked to show that $Y_1 = X_1$ and $Y_2 = X_1 + X_2 - \lfloor X_1 + X_2 \rfloor$ are also independent uniform samples ...
1
vote
1answer
54 views

Find probability from uniform distribution

Let $X$, $Y$ be two independent random variables from $U(0,1)$. Then find $P[Y>(X-1/2)^2]$. I initially tried drawing the figure but that seemed complicated. I then took expectation on both sides ...
0
votes
1answer
276 views

If $X$ and $Y$ are independent Uniform(-1,1), what is the pdf of $\cos(X) \cos(Y)$?

I have two random variables $X$ and $Y$ which are independent and Uniformly distributed in the interval $\left[-\pi, \pi\right]$. What will be the pdf of $\cos\left(X\right)\cos\left(Y\right)$?
0
votes
1answer
781 views

Generating correlated uniform random variable with R [duplicate]

I am trying to generate Correlated Uniform Random Variables with given mean, standard and correlation structure. I have looked through various posts in this topic including this(Can I use the Cholesky-...
0
votes
1answer
58 views

order statistic (random sample from U(0,1) )

for a random sample $X_1 , X_2 .... X_5$ , from a Uniform (0,1) , isn't the distribution of median (say y) be given by $$ f(y)= \frac{y^2 (1-y)^2}{\beta (3,3)}\,,\qquad 0 < y < 1 $$ this can ...
5
votes
1answer
132 views

Finding pdf of transformed variable for uniform distribution

This is from MITx's Intro to Probability and Statistics course, the problem is on this page. Suppose $X \sim \textrm{Uniform}(0,1)$ and $Y=X^3$. Find the pdf for $Y$. Since it's a uniform ...
1
vote
0answers
227 views

Sum of uniformly distributed random variables over different intervals?

Let $\{X_i\}_{i=1}^N$ be $N$ random variables uniformly distributed over the intervals $[a_i, b_i]$ respectively. How does the sum: $$\sum_{i=1}^N X_i$$ distribute? This is a generalization of the ...
3
votes
1answer
96 views

Interval of a transformation of a uniform variable

Suppose we have a uniform random variable $U$ which is defined on the $[0,1]$ interval. Consider the transformation:$$X=k\times \log(U)$$ How would I go about calculating the interval on which $X$...
0
votes
1answer
31 views

Test for RVs with known probabilities?

I have written code that generates a sequence of distinct integers. The integers are assumed to occur in the sequence with fixed probabilities. For example, if the sequence contains the numbers [-1,0,...
10
votes
1answer
894 views

Measure the uniformity of a distribution over weekdays

I have a similar problem to the question asked here: How does one measure the non-uniformity of a distribution? I have a set of probability distributions over the days of the week. I want to measure ...
1
vote
1answer
457 views

Variance of a continuous uniformly distributed random variable

I would like to calculate the variance of a uniformly distributed continuous random variable. The probability density function of a uniformly distributed continuous random variable is $$f_{X}(x) = \...
2
votes
1answer
240 views

Probabilities of conditional expectation values in uniform distribution

Let's consider a continuous random variable $X$ as follows: $f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\ \frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$ ...
8
votes
2answers
273 views

PDF of a sum of dependent variables

This is a direct continuation of my recent question. The thing that I actually want to get is the distribution of $a+d+\sqrt{(a-d)^2+4bc}$, where $a,b,c,d$ are uniform in $[0,1]$. Now, the ...
17
votes
2answers
436 views

What's the distribution of $(a-d)^2+4bc$, where $a,b,c,d$ are uniform distributions?

I have four independent uniformly distributed variables $a,b,c,d$, each in $[0,1]$. I want to calculate the distribution of $(a-d)^2+4bc$. I computed the distribution of $u_2=4bc$ to be $$f_2(u_2)=-\...
2
votes
0answers
432 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
0
votes
2answers
2k views

Probability of finding a point in the unit circle?

Consider the experiment where a pair of numbers (x,y) is chosen at random in the unit square; that is, x and y are uniform (0,1) random variables. What is the probability of (x,y) lying within the ...
3
votes
1answer
118 views

Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$

Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y > X, Y \in [-1,1] \}\,?$$ I managed to compute the ...
18
votes
2answers
2k views

Uniform random variable as sum of two random variables

Taken from Grimmet and Stirzaker: Show that it cannot be the case that $U=X+Y$ where $U$ is uniformly distributed on [0,1] and $X$ and $Y$ are independent and identically distributed. You should not ...
3
votes
1answer
240 views

how calculate expected value

(Ross [2009], p.162) The current in a semiconductor diode is often measured by the Shockley equation I = I0(e^aV-1) where V is the voltage across the diode; I0 is the reverse current; a is a constant; ...
3
votes
1answer
2k views

Measure for the uniformity of a distribution

I can't seem to find a well established and simple statistical measure of uniformity in occurrence datasets in the presence of zero-valued categories. I've looked at Shannon's entropy which seems to ...
0
votes
1answer
676 views

“Convert” Rayleigh random variable into a Uniform random variable?

I have a nested question of sorts. My first question, is that I am wondering if it is possible to 'convert' a Rayleigh random variable into a uniform random variable, and how one may do this. ...
2
votes
2answers
16k views

Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables

Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval $(-1,1)$. Determine the density for $U=\min(X,Y)$ and for $W=\max(X,Y)$
5
votes
1answer
3k views

Derivation of Rayleigh-distributed random variable

I only have a uniform distribution function between [0,1]. And from this distribution, I should generate a sequence of Rayleigh distributed random variable using some software. Anyhow, I was able to ...
29
votes
4answers
20k views

How does one measure the non-uniformity of a distribution?

I'm trying to come up with a metric for measuring non-uniformity of a distribution for an experiment I'm running. I have a random variable that should be uniformly distributed in most cases, and I'd ...
3
votes
2answers
2k views

Problems with extremum of two uniform random variables

Here is the problem from the book: Let $X = \min(U,V)$ and $Y = \max(U,V)$ for independent $\text{uniform}(0,1)$ variables $U$ and $V$. Find the distributions of a) $X$; b) $1-Y$; c) $Y-X$. I ...