Weighted Average of Uniformly Distributed RV [duplicate]
Let $x \sim U[0,1]$ and $y\sim U[0,1]$. Let $z= \omega\, x+ (1-\omega)\,y$, where $\omega\in[0,1]$. The pdf of $z$ is a trapezoidal distribution over $[0,1]$:\begin{equation*}\begin{aligned}f(z)&=...
View Articletransformation of uniform random variables
Let $U_1, U_2,...,U_n$ be a sequence of independent random variables with Uniform distribution over the interval $(0, 1)$ and let $Y = -\frac{1}{\lambda} log(U_1)$ . what is the distribution of Y? i...
View ArticleOn the estimated formula of covariance of two random variables
We define the covariance of two random variables $X$ and $Y$ as $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$. The covariance measures the "linear dependence" between the two r.v s.But in a lot of places, on...
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Why is the CDF of a sample uniformly distributed
I read here that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution.I have...
View ArticleConditional Probability Uniform Bivariate Transformation Distribution
I'm reviewing probability theory from years ago and am a bit rusty. I'm not sure how to calculate the conditional probability for a uniform distribution after a bivariate transformation.Suppose X and Y...
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Asymtotic distribution of the MLE of a Uniform
A property of the Maximum Likelihood Estimator is, that itasymptotically follows a normal distribution if the solution is unique.In case of a continuous Uniform distribution, the Maximum Likelihood...
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What is this hybrid(mixed) random variable’s variance?
X ∼ Uniform(a,b), a<b (Discrete) where f(x)=1/n where n=b-a+1 and Y ∼ Uniform(c,d), c<d (Continuous) where g(y)=1/d-c. X and Y are independent. Let z = x - y. I was able to find the E(Z), however...
View ArticleRecursive Uniform Distribution Expectation Question
Suppose we draw some k ~ Unif(0, 1). Then, we will draw some $u_1$ ~ Unif(0, 1). If $u_1 < k,$ we stop. Else, we will draw $u_2$ ~ Unif(0, $u_1$). We will continue drawing until $u_n < k,$ where...
View ArticleVariations on Uniform Priors with Binomial Signals? [duplicate]
the standard result is that with a uniform prior on p (from 0 to 1) and binomial signals (h H signals and (n-h) L signals from n draws, each with probability p), the posterior mean is (h+1)/(n+2) and...
View ArticleBounding maximum inner product out of $n$ randomly sampled unit norm vectors
Let $w \in \mathbb{R}^d$ have unit norm and $x_1, ..., x_n \in \mathbb{R}^d$ be $n$ randomly sampled vectors from the uniform distribution over the $d$-dimensional unit sphere. Can one obtain a lower...
View ArticleGeneralization of the Irwin-Hall distribution for general linear combinations...
Consider the random variable $Z$, defined by:$$Z = \sum_{k=1}^n c_k X_k$$where $X_k \sim U[0,1]$ is a real random variable with continuous uniform distribution between 0 and 1, and the $c_k$ are real...
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Is it possible to uniformly draw points over a $D-2$ sphere, given that one...
Suppose I have the following scenario:And I am aware of an algorithm to draw uniformly from (in this case) the 2-sphere. Does this same algorithm readily extend to the situation where I randomly take a...
View ArticleJoint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first need...
View ArticleAdvantages of Box-Muller over inverse CDF method for simulating Normal...
In order to simulate a normal distribution from a set of uniform variables, there are several techniques: The Box-Muller algorithm, in which one samples two independent uniform variates on $(0,1)$ and...
View ArticleDistribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d...
Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$.Simulation suggests the...
View ArticleHow can I test the fairness of a d20?
How can I test the fairness of a twenty sided die (d20)? Obviously I would be comparing the distribution of values against a uniform distribution. I vaguely remember using a Chi-square test in college....
View ArticleDiscrete test statistics cannot form a uniform P-value?
I received this feedback on my permutation test design from a collaborator and I'm wondering if his claim is valid. My test statistics are discrete (like counting the number of red marbles found after...
View ArticleIf $20 $ random numbers are selected independently from the interval $(0,1) $...
If $20 $ random numbers are selected independently from the interval$(0,1) $ what is the probability that the sum of these numbers isat least $8$?I tried to take this question...
View ArticleSampling 1 item from groups of correlated values and combining the statistics
We have a dataset consisting of several groups of observations:GroupObjectValueGr_1Ob_1V_1Gr_1Ob_2V_2Gr_2Ob_3V_3.........All values lie in the interval [0,1].In each of the groups Gr_i the values are...
View ArticleBayes Estimate for Mean Squared Loss in Uniform Prior
Can some one please help me out in Verifying if my prior distribution is uniform then will my Bayes estimate will always be MLE or UMVUE?If $X_i$ follow iid $N(\theta,1)$ and prior distribution of...
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