Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
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Joint density of a bounded random vector
Let $X = (X_1, \dots, X_n)$ be a random vector with support $\mathbb{R}^n$, and with distribution $F_X(x_1, \dots, x_n)$ and density $f_X(x_1,\dots, x_n)$. Consider the bounded transformation of $X$ ...
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Creating an Estimator for the Dimension of Bernoulli-distributed Vectors from Observed pairwise Dot Products
I have I individuals defined by vectors $P_i \sim \mathcal{B}(1,1/2)^d$ iid. We can note $\overline{P}_i = \langle P_i, \textbf{1} \rangle$ the proportion of 1's in individual i; $c_{ij} = \langle P_i,...
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Does probability flow ODE trajectory (in the context of diffusion models) represents a bijective mapping between any distribution to a gaussian? [closed]
I have read several papers about diffusion models in the context of deep learning.
especially this one
As explained in the paper, by learning the score function $(\nabla \log(p_t(x)))$, probability ...
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Histograms: discrete data points
I would like some clarification on histograms as I am getting conflicting information.
Question 1
Can histograms be used for discrete data?
I found many sources say that its only used for continuous ...
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Pearson's Chi-Square test, set distribution into categories but desired property within
I have a dataset which falls into 4 categories and I would like to use a Pearson's Chi-square test but I am unsure about the underlying parameters for the Chi-square test statistic. The scenario is as ...
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Calculating expectation by manipulating into the MGF
Given that the moment generating function of the random variable X is
$M_X(t)=\frac{1}{1-2t}$, calculate the expectation of $Y^n,n\in \mathbb{N} $, where
$Y=365 \times 0.3^X$.
Here is my attempt. Use ...
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The distribution of a statistic about normal sample [closed]
$X_1,X_2,...,X_n$ are i.i.d. $N(\mu,\sigma^2)$ samples, $\bar{X}$ and $S^2$ are sample mean and sample variance, then what is the distribution of $Z=\dfrac{X_1-\bar{X}}{S}$? At first I thought it ...
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UMP test for Bernoulli distribution one sample
Given $\mathcal{X}=\{0,1\}$, $\Theta= \{ 1/2,1/4 \}$ and $\mathcal{P}= \{ B(\theta):\theta \in \Theta \}$, I am trying to construct an UMP test of size $\alpha \in (0,1)$ for
$$H_0: \theta=1/4 \,H_1: \...
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Encouraging sparsity at block level or element-wise level?
I have an objective function $f(W)$, where $W$ is a $Kp \times Kp$ matrix. We can view $W$ is a $p \times p$ block matrix, where each block has the dimension $K \times K$. Now to optimize $f(W)$, I ...
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Intuition for bounds of Adaptive Conformal Inference
I have been reading the paper by E. Candès and Gibbs about Adaptive Conformal Inference (here is the original papel). The main idea is to update the miscoverage level $\alpha_t$ as
$
\begin{cases}
\...
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Help needed findint the expected value of this stopping time
Let $\xi_i$ be iid random variables with $\mathbb{E}[\xi_i]=0$, and define:
$$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$
Now, define:
$$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$
How can I find $\...
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Finding the pivot distribution
I am having trouble with this exercise. It is given below.
Let it be X volume sample $n$ from the logistic distribution, with density $f(x;\theta,\sigma)=\frac{e^{-\frac{x-\theta}{\sigma}}}{\sigma(1+e^...
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Does the invariance property hold for consistent estimators within an indicator function?
Let $X_{n}$ denote a sequence of random variables. Then, $X_{n} = c + o_\text{P}(1)$ for some constant $c$ if, for all $\epsilon > 0$, $$\Pr\left(\left|X_{n} - c \right| \geq \epsilon \right) \to 0$...
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Determining the Rao-Cramer Lower Bound for an Unbiased Estimator [closed]
I am having trouble solving the following problem from my exam:
Let (X, Y) be a sample from a distribution with density $f(x, y; \theta)=e^{-\theta x-\frac{y}{\theta}}$, where $x>0$, $y>0$ and $...