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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31 views

The distribution of the sum of a uniform random variable and a binomial random variable

I'm asked to find the distribution of $U=X+Z$, where $X\widetilde~R(0,1)$ - That is, $X$ has a uniform distribution for $x\in]0;1[$ $Z\widetilde~bin(1,1/2)$ - That is, $Z$ has a binomial ...
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0answers
22 views

uniform distribution probability and mle

For part a), isn't the probability = 1? And I'm not sure what happens as $n\rightarrow\infty$; isn't the probability 1 also?
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1answer
48 views

Why does the MLE of the uniform distribution not satisfy a Central Limit Theorem?

For $X ~ U(0,\theta$) The MLE of $\theta = \max{x_i}$. Why does this not satisfy $\sqrt{n*I(\theta)} *( \max(x_i) - \theta) -> Z $ Where Z has a normal distribution? I understand that $\max{x_i} ...
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0answers
41 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
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0answers
41 views

Obtain distribution of mid-range in uniform

I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation. $M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = ...
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1answer
47 views

Conditional distribution on the unit circle and a square

Let (X,Y) be uniformly distributed over $B=\{(x,y) \in \mathbb{R}^2: x^2+y^2 \leq 1 \}$ resp. $Q=[-1,1]^2$. Now I want to calculate the conditional distributions and of Y given X=x. And then the ...
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2answers
45 views

Uniform distribution with unknown interval boundary

Suppose that a random variable $Y$ is uniformly distributed on the interval $[-a,a]$, with $a > 0$. Suppose that the random variable $X$ is uniformly distributed on the (stochastic) interval ...
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1answer
20 views

Calculate $P(A_n)$, where $P$ uniform distribution on $[0,1]$

$P$ uniform distribution on $[0,1]$. $$A_n=\bigcup_{i=1}^{2^n-1} \left [ \frac{2i-1}{2^n}, \frac{2i}{2^n} \right ], n \in \mathbb{N}$$ To calculate $P(A_n)$ do we have to do the following?? ...
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1answer
66 views

Distribution of differences between adjacent sorted uniform random variables on $[0,1]$

I saw this question on Mathematica.stackexchange, and I wonder what distribution the answer gives. Asymmetric definition Let $(X_1,X_2,\ldots,X_{n-1})\sim$ i.i.d. $U[0,1]$, and ...
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0answers
22 views

Universal hash function when size of hash is p^m

Can we define universal hash function from $U \rightarrow T$ when $T=\{0,1,2,..,m-1\}$ and $m=p^a$? (where $p$ is a prime and a is an integer) I know that we can define universal hash funciton when ...
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0answers
107 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
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2answers
58 views

Random variables and Linearity

I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a ...
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1answer
28 views

Number of uniform hash functions

how many uniform hash function I can create when I want to hash elements from $U$ where $|U|=m \cdot r$ , $m,r$ are integers. a hash function $h:U \rightarrow T $ , $|T|=n$ is uniform if ...
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1answer
36 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
4
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2answers
90 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
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3answers
47 views

Biggest among three uniform random variables

Question: Given three random variables $X, Y, Z$ of independent uniform distribution in range [0,1], what's the probability for $X$ to be the biggest one? I've come up with two solutions but they ...
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1answer
37 views

Resultant mean and variance of gaussian distribution

X be random Gaussian variable with mean u1 and variance v1. u1 itself is a random variable which is also gaussian distributed with mean u2 and variance v2. Then the distribution of X will be ...
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0answers
41 views

Finding the limiting distribution $n\min(X_1, \dots , X_n)$ with uniformly distributed $X_i$

Find the limiting distribution of $nY_n$ where $Y_n = \min(X_1, ..., X_n)$ and $X_1, ..., X_n\sim \operatorname{unif}(0,2)$ are uniformly distributed random variables. Here is what I did: ...
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0answers
25 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...
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2answers
57 views

Why is this true? (sum of 2 uniform distributions)

If $X\sim U[0,1]$ and $Y\sim U[-1,0]$ and they are independent, then the distribution of $X+Y$ is not simply $U\sim [-1, 1]$, but it is the sum of 2 independent $U\sim [-0.5 ,0.5]$ distributions. Why ...
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1answer
71 views

Cdf and Pdf of independent random variables(iid)

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
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3answers
38 views

Probability uniform with transformation

Given $X,Y$ being discrete random variables that are independent and can take on values $[0,1,\dots,N]$ with equal probability, what is the distribution of $\max[X,Y]=Z$? Or any other transformation ...
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0answers
20 views

How to rewrite function for squared uniform distribution

The question is as follows: I am evaluating the following integral: $$\int_o^1\frac{\exp(\sqrt{1-x^2})}{\sqrt{x}}dx$$ by assuming it equals $E[f(U)]$ for a uniform distribution. I worked it out via ...
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1answer
52 views

What is the centroid of $x^2+{(-y^3+1)}^{2/3}=1$

I am wondering what exactly is the centroid of $x^2+{(-y^3+1)}^{2/3}=1$. It is a closed implicit shape. I want to know if solving for the centroid is the same thing as solving for a point with the ...
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0answers
11 views

Dinamic programming for sum of two largest values from a Uniform parent

I must find a recursion formula to find the sum of two largest values from a Uniform parent [0,1]. I need dinamic programming but I don't know how to organize it.
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0answers
66 views

Generate random numbers with beta distribution from uniform distribution

How can I generate a series of random numbers with beta distribution from random numbers with uniform distribution? I am aware that using inverse transformation method is at least very difficult or ...
0
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1answer
20 views

Finding a measurable function with an independent uniform distribution

Suppose $X,Y,U$ are random variables on some probability space such that $U$ is independent of $(X,Y)$. Prove there exists a measurable function $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
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2answers
102 views

Expected value of the sum of the two largest values from a Uniform parent

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?
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0answers
94 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
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3answers
36 views

Find PDF of the random variable Y. Please help!

Let $X_1$ and $X_2$ be independent and identically distributed Uniform $(0,1)$ random variables. Let $Y = \max(X_1, X_2)$. Find the PDF of the random variable $Y$. I am having a hard time progressing ...
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0answers
14 views

Even dirstibution of a small set of random choices into a small set of buckets

Is there a way to evenly distribute randomly selected small set of items from a relatively larger set into to a small number of buckets using a hash function? For ex: Randomly select 20 numbers from ...
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0answers
29 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
4
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2answers
220 views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
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65 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...
1
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1answer
29 views

Shuffeled coin tossing

Had a small question: Let's consider the probability space $(\Omega, \mathfrak{F})=([0,1], \mathfrak{B})$ with Lebesgue measure $\mathbb{P}$, $\mathfrak{B}$ is Borel sigma algebra. Lets expand a ...
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2answers
198 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
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1answer
58 views

probability: nonlinear best predictor $\hat{Y} = g(X)$

Consider $X\sim\mathcal{U}(-1,1)$ and $Y = X^2$. The nonlinear predictor is defined as $$ \hat{Y} = g(X) = E_{Y|X}[Y|x_i] $$ Now $E_{Y|X}[Y|x_i] = \int_{-\infty}^{\infty}y\frac{f_{X, Y}(x, ...
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1answer
31 views

probability: best linear predictor $\hat{Y} = aX + b$

Let $X\sim\mathcal{U}(-1, 1)$ and $Y = X^2$. Since the best linear predictor is defined as $$ \hat{Y} = E_Y[Y] + \frac{\text{cov}(X, Y)}{\text{var}(X)}(x - E_X[X]) $$ Can I simple just write it as ...
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1answer
52 views

Determine probability of fewer than a certain number of events

Could anyone help with the following problem? My guts is telling me that the answer to part (a) is a normal distribution. Mainly, because I can't see where a uniform distribution would fit in this ...
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1answer
89 views

continuous probability: signal to noise ratio $\mu^2/\sigma^2$

$\DeclareMathOperator{\var}{var}\DeclareMathOperator{\cov}{cov}$ The signal-to-noise ratio (SNR) of a random variable quantifies the accuracy of a measurement of a physical quantity. It is defined ...
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1answer
80 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...
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39 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
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1answer
65 views

Find percentage of conforming items for normal and uniform distributions

I'm given the following problem: Could anyone give some insight into how to solve this? I understand that for the normal distribution, I will likely have to look the percentage up in a table. ...
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2answers
90 views

assume that X and Y are independent with X ~ UNIF(-1,1) and Y~UNIF(0,1).

I am trying to find the probability that the roots of the equation h(t)=0 are real, where h(t)=t^2+2Xt+Y of the given data. I know that I need to look at Uniform continuous distributions but I am ...
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2answers
58 views

Uniform Probability Distribution 1

A manager of a department store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 2 to 4 minutes. If it takes the elevator 30 ...
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1answer
42 views

How can I prove that Xn converges to 0 in probability?

Let $X_n\sim U[-1/n,1/n]$. Since for convergence in probability for every $\epsilon>0$, $$ \lim_{n\to\infty} P(|X_n - X|\ge \epsilon) = 0 $$ Hence, $P(|X_n-0|\ge ...
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1answer
50 views

Expectation of continuous uniform distribution

I'm having a problem with a basic probability problem. There is a stick which is 4 units in length, we break it in two pieces and the breaking point is randomly distributed. After this we form a ...
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2answers
232 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
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1answer
39 views

Conditional probability explained

Sorry for the dumb question, but it seems that I'm missing something pretty straightforward Abstract Suppose you are throwing one cube of dice, and you have thrown value "6" ten times in a row, ...
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1answer
66 views

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]? $E[e^{\frac{2X}{3}} - 3] = \int_0^2 \! e^{\frac{2X}{3}} - 3 \, \mathrm{d}x$ $= \frac{3}{2}(e^{\frac{4}{3}} - 5) = -1.8095$ I am integrating over the ...