Advantages 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 ArticleWhy would predicted values be normally-distributed when the actual values are...
I'm building a supervised learning model where the target variable is a uniformly-distributed continuous value ranging from 0 to 1 (originally a rank value from 1 to 38000, then scaled down to 0-1)....
View ArticleDistribution of $X^3/Y$ where $X$ and $Y$ are uniformly distributed
$X$ and $Y$ are two uniformly distributed variables in interval $(a,b)$ and $(c,d)$ respectively ($a>0$ and $c>0$).What is the distribution of the variable $Z$ defined as $Z=X^3/Y$?
View ArticleDistribution function of $1/X$ when $X$ is uniform on $[-1,1]$
(from The Probability Tutoring Book, C. Ash, p. 157)Find the density of $Y$ if $Y = 1/X$ and $X$ is uniform on $[-1,1]$.The distribution function given in the answer key is$$F(y) =...
View ArticleConditional Probability Uniform Bivariate Transformation Distribution
I'm reviewing probability theory from years ago and am a bit rusty. I'm not sure how to calculate the conditional probability for a uniform distribution after a bivariate transformation.Suppose $X$ and...
View ArticleDetermining the range of a random distribution from a finite set of numbers...
My problem is as follows: I am provided a set of 5 integers that have been randomly generated from the range of 1 to n. Using these 5 integers, I have to determine the original n. Obviously, this...
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How do you uniformly sample spans from a bounded line?
Suppose you have a bounded and continuous line. For example, the line could include all real numbers between 0 and 3. How do you sample spans from the line such that...Any point on the line has an...
View ArticleGLMM model choice and uniformity questions [closed]
I am trying to find the best way to deal with my data. I basically have a large dataset with many plant species originating from many different provenances (each species has at least 4). I have...
View ArticleProbability Density of the Sum of Two Un-identical Uniform Random Variables
Let $X$ ~ Uniform$[a,b]$ and $Y$ ~ Uniform$[c,d],$ where $a\le b\le c\le d.$ Find the probability density of $Z = X + Y.$I know I have to use the convolution formula$$f_Z(z) = \int_{-\infty}^\infty...
View ArticleDifferentiability issues at the boundary for parameter-dependent support
Differentiability of $\log p_{\theta}(x)$ at the boundary for $\mathrm{Uniform}(0,\theta)$Let $X_1, \cdots, X_n$ be i.i.d. with density$$p_{\theta}(x) = \frac{1}{\theta}, \quad 0 \leq x \leq...
View ArticleHow to derive an estimator for the parameter of a continuous uniform...
$X_1, X_2,\dots.,X_n$ are i.i.d. random variates drawn from a continuous uniform distribution over $[0,\theta].$ The sufficient statistic is denoted $\max$.I want an estimator $e$ of $\theta$ that...
View Articlehow to write the joint density of two correlated uniform random variables?
Suppose the marginal distributions of $X$ and $Y$ are both $U[a,b]$, where $U[a,b]$ denotes uniform distribution on $[a,b]$. When $X$ and $Y$ are correlated, how to write their joint density for some...
View ArticleExponential family definition appears vacuous
I am going through Michael Jordan's notes on exponential families and an exponential family is defined w.r.t. functions $h(\cdot), T(\cdot)$, and parameter $\eta$ such that$$p(x | \eta) = h(x)...
View ArticleWhat's the distribution of the closest point from uniform samples?
Suppose you have $N$ values $x_1, \ldots, x_N$ that are uniformly sampled in $[0; 1]$. For a random $x_k$ amongst the $(x_i)_i$ (with equiprobability), what is the expected value of the distance...
View ArticleRegression where the conditional distribution is uniform (limiting behavior...
I am reading about the power exponential distribution a.k.a. Subottin distribution a.k.a. "generalized normal distribution" with pdf:$$ f\left(y | \mu, \sigma, \kappa\right) =...
View ArticleIs there something like a uniform distribution bounded on two intervals (or a...
Let's say that I want to have a distribution from which there is an equal probability to draw a number between -0.9 and -0.3 and also a number between 0.2 and 0.8. Is there a distribution which allows...
View ArticleWhat posterior for exponential distribution with uniform prior?
Suppose I have the PDF of an exponential distribution which $\lambda$ parameter is unknown:$$f(x\mid\lambda)=\lambda \, e^{-\lambda x} \quad \lambda>0 \quad \text{support} = \mathbb{R_+^*}$$Suppose...
View ArticleJeffreys prior for continuous uniform distribution
A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by:$f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ is...
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Effectively Visualizing P-value Distributions with Excessive 1.0 Values
I'm dealing with a discrete statistical test scenario where a significant portion of the P-values calculated are exactly 1.0, due to the nature of the test statistic not exceeding its observed value in...
View ArticleFind a two dimensional sufficient statistic for $\theta$
Let $\{X_i\}_{i=1}^n$ be conditional independent given $\theta$ with distribution$$p_{X_i | \theta} (x |\theta) = \frac{1}{2i\theta}, \ -i\theta<x<i\theta.$$Find a two dimensional sufficient...
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